Higgs Properties at the LHC : Implications for the Standard Model and for Cosmology 978-3-319-63402-9, 331963402X, 978-3-319-63401-2

Table of contents :
Front Matter ....Pages i-xviii
Introduction—Realisation of the EW Symmetry in the SM (Jason Tsz Shing Yue)....Pages 1-23
Spin Determination of the LHC Higgs-Like Resonance (Jason Tsz Shing Yue)....Pages 25-40
Probing \(\mathcal {CP}\)-violating Top-Yukawa Couplings at the LHC (Jason Tsz Shing Yue)....Pages 41-73
Electroweak Phase Transition and Baryogenesis (Jason Tsz Shing Yue)....Pages 75-107
Conclusions (Jason Tsz Shing Yue)....Pages 109-111
Back Matter ....Pages 113-130

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Springer Theses Recognizing Outstanding Ph.D. Research

Jason Tsz Shing Yue

Higgs Properties at the LHC Implications for the Standard Model and for Cosmology

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

Theses are accepted into the series by invited nomination only and must fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder. • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to scientists not expert in that particular field.

More information about this series at http://www.springer.com/series/8790

Jason Tsz Shing Yue

Higgs Properties at the LHC Implications for the Standard Model and for Cosmology Doctoral Thesis accepted by The University of Sydney, NSW, Australia

123

Supervisor Prof. Archil Kobakhidze The University of Sydney Sydney, NSW Australia

Author Dr. Jason Tsz Shing Yue Department of Physics National Taiwan Normal University Taipei Taiwan

ISSN 2190-5053 Springer Theses ISBN 978-3-319-63401-2 DOI 10.1007/978-3-319-63402-9

ISSN 2190-5061

(electronic)

ISBN 978-3-319-63402-9

(eBook)

Library of Congress Control Number: 2017948609 © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Supervisor’s Foreword

The groundbreaking discovery of the Higgs boson particle at the CERN Large Hadron Collider (LHC) found fresh physics graduate Jason Yue at the start of his Ph.D. studies in the University of Sydney. This work is a result of his research in one of the most fascinating areas of fundamental physics. The thesis starts out with a concise introduction to basic theoretical aspects of the Standard Model of particle physics, including discussion on effective field theories and renormalisation, unitarity, gauge invariance and the Higgs mechanism and nonlinear realisation of the electroweak symmetry. In Chap. 2, the author discusses possible spin and parity assignments for the LHC resonance. The determination of these quantum numbers is an important experimental task for establishing the Higgs mechanism and the related Higgs particle. By utilising theoretical arguments on perturbative unitary and electroweak precision measurements, the author established that generic Higgs impostors with spin-2 and even and odd parities are excluded, leaving the room only for the spin-0 Higgs particle. In view of the fact that spin-2 Higgs impostors with generic interactions are notoriously difficult to exclude by the standard experimental analysis, this result is of a significant importance. In Chap. 3, Jason analyses anomalous top-quark Higgs Yukawa couplings within the framework of nonlinearly realised electroweak symmetry. Using the collider data, constraints on CP-violating couplings are obtained, and prospects of their measurements in future experiments have been elucidated. Interactions of the Higgs boson with the heaviest standard model particle, the top-quark as well as Higgs self-interactions may play a very important role in the very early universe. In Chap. 4, Jason studied cosmological implications of the model with anomalous Higgs couplings. He established an intriguing connection between Higgs trilinear coupling and the nature of the electroweak phase transition and the dynamical generation of the matter–antimatter asymmetry in the universe.

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Supervisor’s Foreword

The topics studied in Jason Yue’s thesis are in the focus of worldwide efforts of experimental and theoretical particle physics communities. I believe this work will be useful for young Ph.D. students as well as experienced researchers. Sydney, Australia June 2017

Archil Kobakhidze Associate Professor

Abstract

The aim of this thesis is to study the properties of the 125 GeV Higgs-like resonance discovered at the Large Hadron Collider (LHC) in 2012 and to elucidate its role in electroweak (EW) symmetry breaking. The first step is to study the spin and charge parity (J CP ) assignments for this resonance, which are alternate to the Standard Model (SM) Higgs. In particular, we use unitarity arguments to eliminate the possibility that the new resonance is a spin-2 impostor. Furthermore, it was found that such an impostor leads to large deviations from the observed oblique precision parameters. This resonance must then be a scalar, a pseudoscalar or a mixture of these two cases. A nonlinearly realised electroweak symmetry may lead to CP-violating top-Yukawa couplings. Collider data was used to put indirect constraints on the modulus, yt and CP-phase, n, which parameterise such couplings. We then studied the LHC potential to probe the tth coupling directly through the pp ! thj channel, focusing on the scalar (jnj ¼ 0), pseudoscalar (jnj ¼ 0:5p) and maximally mixed (jnj ¼ 0:25p) scenarios. It was found that large QCD backgrounds in h ! bb decays make it difficult to observe thj production. Instead, we demonstrated that higher signal significance is expected in h ! cc decays, where signal reconstruction is significantly improved due to cleaner signatures. The lepton forward– backward asymmetry was found to be a good CP-observable. As it measures the polarisation of the produced t-quark, it allows different n’s to be distinguished. The last part of this thesis examines the phase transition (PT) of a nonlinearly realised EW gauge symmetry. Electroweak baryogenesis may then generate the observed matter–antimatter asymmetry without augmenting the SM particle content. This is realised by extra sources of CP-violation and first-order PT due to the anomalous top-Higgs and cubic Higgs interactions of the non-standard gauge structure.

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Preface—The Higgs Discovery

Thus invariance principles [of symmetry] provide a structure and coherence to the laws of nature just as the laws of nature provide a structure and coherence to the set of events. —D.J. Gross [1]

The discovery of a Higgs resonance, hð125Þ1, was announced by ATLAS and CMS in 2012 [2, 3]. In the Standard Model (SM) of particle physics, the Higgs boson plays an instrumental role in giving masses to the particles whilst keeping gauge invariance [4–8]. This reconciliation with gauge symmetry principles is an important step in successfully describing the fundamental particles and their interactions. The Higgs discovery follows from analysing the 5 fb1 and 20 fb1 of data collected in 7 and 8 TeV pp-collisions respectively. This was mainly driven by the h ! cc and h ! ZZ  decay channels where they individually reached significances of [ 4r and [ 5r respectively. The same resonance was identified with [ 2r excess in the h ! WW  channel within the same mass region, but with a lower resolution. The combination of these channels resulted in a local excess of [ 5r, suggesting that a random statistical fluctuation is unlikely2. The high mass resolution in the h ! cc; ZZ  channels was exploited by the collaborations to yield the first combined measurement of the Higgs mass [9] (cf. Fig. 1): mh ¼ 125:09  0:21ðstat:Þ  0:11ðsyst:Þ GeV: In the forthcoming years, the focus of the community is to pin down the role that hð125Þ plays in the electroweak (EW) symmetry breaking. The first step in characterising the Higgs-like resonance is to establish its spin and CP-properties (J CP ), 1

We will subsequently use this interchangeably with h to denote the resonance. A 5r significance corresponds to a probability or p-value of  107 assigned to obtaining the current data without the resonance.

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Preface—The Higgs Discovery ATLAS and CMS LHC Run 1

Total

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Fig. 1 Combination of the ATLAS and CMS measurements on the mass of the Higgs-like resonance using the h ! cc and h ! ZZ  decay channels. Source [9]

which are the quantum numbers3 dictating the Lorentz structure of the possible interactions. This is the subject of our work in [10–12]. The related cosmological implications of such a resonance were subsequently explored in [13]. This thesis will be devoted to explaining this series of works. In Chap. 1, we review the role that the Higgs boson plays in the context of gauge invariance, unitarity and renormalisability. In particular, the Higgsless theory is well described by an effective chiral theory where the electroweak symmetry SUð2ÞL  Uð1ÞY is nonlinearly realised. A scalar is required to unitarise the perturbative scattering amplitudes whilst retaining the perturbative regime, whereas the unitarisation by vector or tensor resonances will eventually lead to a strongly coupled theory4. Also, spontaneous symmetry breaking with a vector or tensor resonance should lead to a vacuum that violates Lorentz symmetry. Current experimental data is consistent with a SM scalar of even parity P (J ¼ 0 þ ). The J ¼ 1 case can be eliminated by the Landau–Yang theorem, but exclusion limits on the J ¼ 2 alternative hypothesis are based on minimal graviton-like couplings. Although the discovery was made solely in the diboson channels, the existence of such scalar couplings to massive bosons is taken as an indication that the minimal Higgs mechanism indeed operates in the SM gauge sector. In recognition of this contribution, Higgs and Englert shared the Nobel Prize in 2013.

Here J refers to spin, and C to charge conjugation, where the particle is exchanged for its antiparticle; P refers to spatial inversion of the particle wavefunction. The eigenvalues associated with these two operators may be either positive or negative. 4 Although it was shown in [14, 15] that a heavy vector state can replace the Higgs in unitarising the scattering of the weak gauge bosons, but this is only so up to a cut-off of K  3 TeV. A more massive scalar or an infinite tower of vector-like resonances (e.g. in extra dimensions or composite scenarios, possibly related via the AdS/CFT conjecture) will then be required to unitarise such theory (cf. e.g. [16, 17]). 3

Preface—The Higgs Discovery

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m κ F vF or

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m κ V vV

Chapter 2 will correspond to our work [10], where we give a theoretical argument against a spin-2 impostor with generic couplings. This is based on the fact that such a resonance should lead to unitarity violation if it is to mimic the SM Higgs decay rate to weak bosons and if no further particles are found. The electroweak precision observables are also shown to be incompatible with such a replacement of the SM Higgs with the impostor. Having ruled out the J ¼ 1 and J ¼ 2 scenarios, one can focus on distinguishing a pure scalar hypothesis against a pure pseudoscalar hypothesis. The preference of the 0 þ case over the 0 case is not unexpected, given that the resonance discovery is made in the diboson modes. Pseudoscalar couplings to massive vector bosons are loop suppress relative to the tree-level scalar mass terms induced by the Higgs mechanism (cf. Chap. 3). As such, the fermion sector should provide a more democratic probe to the CP-structure of the new resonance. In order to show that the minimal Higgs mechanism is also operative in the Yukawa sector, one has to first verify that the Higgs boson couples proportionally to the masses of the fermions, which is required to retain the SUð2ÞL structure in the SM. Global fits where the Higgs couplings to the bosons and fermions are allowed to scale from those in the SM by respective constants ji , reveal that this is indeed the case (cf. Fig. 2). Subsequently, there will be a preference to decay into heavy fermions. Although the top-quark mass is the largest of the fermions, on-shell h ! tt decays are forbidden since mh \2mt . The most favourable fermion decay channels are then h ! bb and h ! ss but no direct fermion coupling could yet be established. There are only evidence for the ss mode at ATLAS (4:5r) [19] and CMS (3:2r) [20]. The dominant Higgs production and decay mode have also been measured to be consistent with the SM prediction. This is evident in Fig. 3, where li parameterises

1.4 1.2 1 0.8 0.6 0.4 0.2 0

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Preface—The Higgs Discovery ATLAS and CMS Preliminary LHC Run 1

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Fig. 3 ATLAS and CMS measurements of the Higgs production (left) and decay (right) rate, as measured with the normalisation li with respect the SM prediction. Source [18]

the normalisation of the measured rate with respect to that of the SM. As gluon fusion production (gg ! h) and diphoton decay (h ! cc) are mediated predominantly by t-quark loops in the SM, the consistency of the data with lggF ¼ lcc ¼ 1 hint at the existence of the top-Yukawa coupling. Assuming a scalar hypothesis, current measurements of the Higgs couplings still allow departures from a linear electroweak gauge structure. As such, the new resonance can be a singlet under the nonlinearly realised electroweak symmetry5. An important ramification is that the singlet is possibly not a CP-eigenstate. Chapter 3 is devoted to explaining [11, 12], which focuses on the collider consequences of a CP-violating top-Yukawa coupling. Information about such a sector may then be obtained from the decay and production rates, as well as kinematic variables. In particular, a global fit of the modulus and CP-violating phase on the anomalous top-Higgs coupling is included. Subsequently, the polarisation of the top-quark in the pp ! thj channel can be inferred from its decay products. Finally, there are long-standing observations which are not addressed in the SM, namely (i) the baryon asymmetry in the universe, (ii) the neutrino masses and mixings, (iii) dark matter, (iv) gravitational interactions, and (v) the stability of the electroweak scale and Higgs mass. Chapter 4 follows our work [13], where we address the issue of baryogenesis. We study the phase transition within the effective field theory of the nonlinearly realised EW gauge group and explain how the

5

Although one can also approach to explain the deviations using higher dimensional operators from the SM effective field theory.

Preface—The Higgs Discovery

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observed baryon asymmetry is accommodated. This is achieved via extra sources of CP-violation from the top-Higgs sector, which together with anomalous cubic Higgs couplings drives a strongly first-order phase transition. The conclusion and outlook is finally presented in Chap. 5. Taipei, Taiwan

Dr. Jason Tsz Shing Yue

References 1. D.J. Gross, The role of symmetry in fundamental physics. Proc. Nat. Acad. Sci. 93, 14256– 14259 (1996). [http://www.pnas.org/content/93/25/14256.full.pdf] 2. ATLAS collaboration, G. Aad et al., Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Phys. Lett. B716, 1–29 (2012). arXiv:1207.7214 3. CMS collaboration, S. Chatrchyan et al., Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. Phys. Lett. B716, 30–61 (2012). arXiv:1207.7235 4. P.W. Higgs, Broken Symmetries and the Masses of Gauge Bosons. Phys. Rev. Lett. 13, 508–509 (1964) 5. F. Englert, R. Brout, Broken Symmetry and the Mass of Gauge Vector Mesons. Phys. Rev. Lett. 13, 321–323 (1964) 6. T.W.B. Kibble, Symmetry breaking in nonAbelian gauge theories. Phys. Rev. 155, 1554– 1561 (1967) 7. G.S. Guralnik, C.R. Hagen, T.W.B. Kibble, Global Conservation Laws and Massless Particles. Phys. Rev. Lett. 13, 585–587 (1964) 8. P.W. Higgs, Spontaneous Symmetry Breakdown without Massless Bosons. Phys. Rev. 145, 1156–1163 (1966) 9. ATLAS, CMS collaboration, G. Aad et al., Combined Measurement of the Higgs Boson Mass pffiffi in pp Collisions at s ¼ 7 and 8 TeV with the ATLAS and CMS Experiments. Phys. Rev. Lett. 114, 191803 (2015). [1503.07589] 10. A. Kobakhidze, J. Yue, Excluding a Generic Spin-2 Higgs Impostor. Phys. Lett. B727, 456– 460 (2013). arXiv:1310.0151 11. A. Kobakhidze, L. Wu, J. Yue, Anomalous Top-Higgs Couplings and Top Polarisation in Single Top and Higgs Associated Production at the LHC. JHEP 10, 100 (2014). arXiv:1406.1961 12. J. Yue, Enhanced thj signal at the LHC with h ! cc decay and CP-violating top-Higgs coupling. Phys. Lett. B744, 131–136 (2015). arXiv:1410.2701 13. A. Kobakhidze, L. Wu, J. Yue, Electroweak Baryogenesis with Anomalous Higgs Couplings, JHEP 04, 11 (2016). arXiv:1512.08922 14. D. Bertolini, Heavy vectors in Higgsless models, Master’s thesis, Università degli Studi di Perugia, 2008 15. R. Barbieri, G. Isidori, V. S. Rychkov, E. Trincherini, Heavy Vectors in Higgs-less models. Phys. Rev. D78, 036012 (2008). arXiv:0806.1624. 89 16. C. Csaki, C. Grojean, H. Murayama, L. Pilo and J. Terning, Gauge theories on an interval: Unitarity without a Higgs. Phys. Rev. D69, 055006 (2004). arXiv:hep-ph/0305237 17. C. Csaki, C. Grojean, L. Pilo, J. Terning, Towards a realistic model of Higgsless electroweak symmetry breaking. Phys. Rev. Lett. 92, 101802 (2004). arXiv:hep-ph/0308038 18. ATLAS, CMS, Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC pp collision data at pffiffi s ¼ 7 and 8 TeV, 2015

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19. ATLAS collaboration, G. Aad et al., Evidence for the Higgs-boson Yukawa coupling to tau leptons with the ATLAS detector. JHEP 04 (2015) 117, arXiv:1501.04943 20. CMS collaboration, S. Chatrchyan et al., Evidence for the 125 GeV Higgs boson decaying to a pair of s leptons. JHEP 05, 104 (2014). arXiv:1401.5041

Acknowledgements

I am most indebted to my advisor Archil Kobakhidze, without whom this series of works is not possible. I thank him for introducing the physics topics to me as well as sending me to various schools and conferences. My graititude also goes towards my co-supervisor Lei Wu, who has helped me in many aspects of the phenomenological works. I am also grateful for the experienced advices that I have received from Michael Schmidt and Kristian McDonald during my studies. A special thanks to the residents of Room 342—particularly Neil Barrie, Adrian Manning, Suntharan Arunasalam, Cyril Lagger and Carl Suster for the countless discussions on physics and other matters. I should also thank the particle physics centre CoEPP and the Australian Research Council for supporting this research. I will cherish the many memories I share with my friends during my Ph.D. studies—especially those with Eric Lee, Marcello Solomon and Angelica Lau. I am also thankful to my unlce Valen, for his encouragments. Last but not least, I would like to thank my family—Eric, May and Jimmy, for their love and support during my academic endeavours.

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Contents

1 Introduction—Realisation of the EW Symmetry in the SM . . . . 1.1 Renormalisability, Unitarity, Gauge Invariances and all that. . 1.1.1 Renormalisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 SM from Gauge Invariance . . . . . . . . . . . . . . . . . . . . . 1.2 Spontaneous Symmetry Breaking and the Higgs Mechanism . 1.2.1 Non-linear Realisation . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 STU Precision Parameters . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Higgs as Singlet Addition from Unitarity Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Spin Determination of the LHC Higgs-Like Resonance . . 2.1 Excluding J ¼ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Massive Spin-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Couplings to Matter . . . . . . . . . . . . . . . . . . . . . 2.3 hZ ! hZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 STU Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Probing CP-violating Top-Yukawa Couplings at the LHC . . . . 3.1 Non-linear Realisation in the Top-Higgs Sector . . . . . . . . . . . 3.1.1 Contribution to Loops . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Bounds on CP-violating Couplings. . . . . . . . . . . . . . . . . . . . . 3.2.1 Branching Ratios and Production Cross Sections . . . . 3.2.2 EDM Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Polarisation Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Lepton Spin-Correlation . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Single Top Production . . . . . . . . . . . . . . . . . . . . . . . .

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3.4 Higgs Associated with Single Top Production at the LHC . . . 3.4.1 Collider Physics at the LHC . . . . . . . . . . . . . . . . . . . . 3.4.2 Observability and Lepton Forward-Backward Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Electroweak Phase Transition and Baryogenesis . . . . . . . . . . . . 4.1 Problems with Electroweak Baryogenesis . . . . . . . . . . . . . . . . 4.2 EW Phase Transition and Effective Potentials . . . . . . . . . . . . 4.2.1 Tree Level Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 One Loop Quantum Corrections . . . . . . . . . . . . . . . . . 4.2.3 Finite Temperature Corrections . . . . . . . . . . . . . . . . . . 4.2.4 Combining Thermal and Quantum Effects . . . . . . . . . 4.3 Bubble Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 B-violation with Sphalerons . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Charge Transport Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Source Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Number Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Interaction Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Approximations to Solve the Transport Equations . . . 4.6 Conversion of nL into nB by Weak Sphalerons . . . . . . . . . . . . 4.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Chapter 1

Introduction—Realisation of the EW Symmetry in the SM

…quantum field theory itself has no content beyond analyticity, unitarity, cluster decomposition, and symmetry —S. Weinberg [1]

In this chapter, we first construct the Higgsless SM as an effective field theory. This is done in Sect. 1.1 using the notions of renormalisability, unitarity and gauge invariance. In Sect. 1.2, unitarity arguments motivates the existence of the Higgs as a minimal scenario. In the same section, we introduce differences between a linearly realised and non-linearly realised electroweak symmetry and argue why the latter is a more generic scenario which should be used to interpret LHC data.

1.1 Renormalisability, Unitarity, Gauge Invariances and all that It is widely believed that the SM is an incomplete theory of nature, valid up to some high energy scale U V . It therefore makes sense to build an effective field theory (EFT) to describe phenomena with energy sufficiently lower than an ultraviolet cutoff at U V . Only effective degrees of freedom relevant to the low energy sector are retained, while the dynamics of those with frequencies higher than U V are coarse-grained. The remnant of the theory when the UV modes are integrated out is an infinite tower of local operators. These operators can be ordered as an operator product expansion, according to increasing powers of E/. This property is ensured

© Springer International Publishing AG 2017 J.T.S. Yue, Higgs Properties at the LHC, Springer Theses, DOI 10.1007/978-3-319-63402-9_1

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2

1 Introduction—Realisation of the EW Symmetry in the SM

by the Appelquist-Carazzone decoupling theorem [2]1 . One can take two different approaches in constructing EFTs, depending on the nature of the problem: • In the top-down approach, the UV complete model (or another effective field theory valid at higher energies) is known. The coarse-graining is to be done in a Wilsonian sense where the field is decomposed into low- (φ L ) and high- (φ H ) frequency modes: φ = θ(|k| − )φ H (k) + θ( − |k|)φ L (k).

(1.1)

The higher frequency modes are integrated out in the path integral of the original action S:  (1.2) ei S [φL ] = Dφ H ei S[φL ,φ H ] , so that the effective action S becomes non-local at the scale of . This can be understood by the fact that heavy particle exchanges are replaced by nonrenormalisable, local operators, as is exemplified by expanding the two-point correlator of a scalar φ:  0|φ(x)φ(y)|0 =

d 4 k eik(x−y) = (2π)4 k 2 + m 2



1  + 4 + ··· 2 m m



δ (4) (x − y). (1.3)

The coarse-grained action in (1.2) then takes the following form:  S [φ] =



  gi () (r +s )/2 1 μ i i Z  ∂ φ∂μ φ + d x Z Ori si (φ, ∂φ) , 2 di −4  i 

 4

(1.4)

:=L

where the operator Ori si is of the schematic form (∂φ)ri φsi . Here, Z  gives the field renormalisation factor which is used to renormalise the infinities of the theory (this notion will be explained below); gi () corresponds to the renormalised coupling constants where the effects of physics at higher energies are encoded in the  dependence. • In the bottom-up approach, the underlying physics at high energies are not known (or the theory is known but strongly coupled). Symmetry and naturalness become important guiding principles in the construction of a Lagrangian (cf. e.g. [1, 5, 6]). In essence, one almost always follows this approach in phenomenological studies. By taking effective Lagrangians of the form in (1.4), one tries to infer the fundamental theory by making measurements and prediction of the coupling 1 The situation is different in theories where spontaneous symmetry breaking is the sole mechanism

for mass generation. In such case m = gφ, and to get large masses, either (i) φ → ∞ and all particles become heavy or (ii) g/m = 1/φ is kept fixed and non-decoupling is observed (cf. e.g. [3], and [4] for a discussion specific to SM Higgs.

1.1 Renormalisability, Unitarity, Gauge Invariances and all that

3

constants gi () at different energies. This approach is then limited by the energy reach of the experiments.

1.1.1 Renormalisability Renormalisability is an indicator for a theory to be UV complete and has been formulated rigorously [7–10]. It prevents operators with mass dimensions > 4 from appearing in the Lagrangian. Since the number of parameters appearing in the Lagrangian is then finite, once these parameters are fixed by measurements of the same number, it is indisputable that consistent predictions can be extrapolated at any perturbative order and scale. Quantum corrections to the bare parameters—masses, coupling constants and field normalisations, result in universal UV infinities that can be reabsorbed as renormalised parameters, which become finite but the trade-off is that these parameters inherit a dependence on the renormalisation scale [11, 12]. However, renormalisability is not a fundamental requirement—EFTs are constructed as quantum field theories (QFTs) where terms are allowed in the action so long it is consistent with symmetries of the theory. EFTs are still able to make useful physical predictions without the restrictions of renormalisability. If there are non-renormalisable terms in the EFT, the infinites cannot be removed by a finite number of counter-terms, instead introducing new ones at each order. Even though power-counting of the weight of E/ associated with each operator (cf. e.g. [12–14]) is applicable at tree level, trouble enters when one considers loop effects of these non-renormalisable operators—let us take the example in [15]:

λ 1 c6 6 1 c8 4 1 φ − φ ∂μ φ∂ μ φ , L ⊃ − φ( + m 2 )φ − φ6 − 2 4! 6!  2! · 4! 

(1.5)

and consider the 1-loop corrections to λ. In a cutoff scheme, these are given by: δλcutoff δλcutoff

 c6 ⊃ 2   c8 ⊃ 8 





d 4k 1 c6 2 ∼ , (2π)4 k 2 − m 2 2 16π 2 d 4k k2 c8 4 ∼ , (2π)4 k 2 − m 2 4 16π 2

(1.6)

which lead to δλ ∼ O(1) contributions for both cases when c6,8 ∼ O(1). However, this is not to say that predictivity is completely lost. This is made clear by employing a mass-independent regularisation scheme (e.g. dimensional regularisation) where the loop contributions become:

4

1 Introduction—Realisation of the EW Symmetry in the SM

δλdim. reg. δλdim. reg.

  2   m 1 c6 μ2 c6 m 2 1 d 4− k , − ln ⊃ ∼ 2 2 (2π)4− k 2 − m 2  16π 2  μ2     k2 c8 μ2 c8 m 2 1 d 4− k m2 . ⊃ ∼ − ln 4 (2π)4− k 2 − m 2 4 16π 2  μ2

(1.7)

By truncating the series up to dimension d, one can still make predictions with accuracy up to O (E/)d−4 . Furthermore, it was shown that if one includes all possible terms consistent with the symmetry of the theory, UV divergences will be proportional to polynomials in momenta and can be completely removed at any given order [16].

1.1.2 Unitarity A closely related concept is unitarity, which is a statement of probability conservation. In the 1960’s there was a serious programme to work out implications of analyticity and unitarity of the S matrix (for representative works, cf. [17, 18]). As related to the opening quote of this chapter, it still remains a conjecture that S-matrices of asymptotic states exhibit well-behaved qualities, such as unitarity and preservation of the symmetries of the Lagrangian [19]. Yet if one accepts this, the corresponding partial wave amplitudes should satisfy perturbative unitarity bounds2 , which we show in Appendix A.1. One observes that non-renormalisable operators are associated with negative mass dimension and may lead to violation of unitarity bounds at high energies. We remark that the occurrence of such violation is implicitly assumed at the tree-level perturbation of the EFT (and so is denoted perturbative unitarity) and that unitarity should be restored in the full theory including all orders of perturbation. Indeed, strongly coupled theories which are non-renormalisable have been shown to be able to self-unitarise [22–24]. In fact, the equivalence between perturbative unitarity and renormalisability have been demonstrated in gauge theories [25–27] and for gravity [28]. We refer the interested reader to recent works such as [29] for further discussion on this topic. The breakdown of perturbative unitarity violation has been used in a sense of ‘nolose’ theorem which guarantees the existence of new physics [30]. One may in fact build the SM by adding new degrees of freedom every time perturbative unitarity is violated [31, 32]. An example is Fermi’s effective theory of weak interaction, which has enjoyed reasonable success via the use of non-renormalisable four-fermion operators. At E ∼ 4πv, where v = 246 GeV is the Higgs vacuum expectation value, the theory enters a strongly coupled regime and the hierarchy established by powercounting of E/ is no longer present. After the introduction of the massive elec-

2 The

first of this type was obtained by Froissart in [20] and recently improved [21].

1.1 Renormalisability, Unitarity, Gauge Invariances and all that

5

troweak bosons to unitarise these perturbative amplitudes, the theory once again runs into further unitarity problems—this time due to the scattering of the longitudinal modes of the gauge bosons. As will be explained in Sect. 1.2, the Higgs boson the minimal addition which restores perturbative unitarity.

1.1.3 SM from Gauge Invariance The idea that symmetry dictates the form of interactions was taken with much faith after Weyl related conservation of electric charge with a U (1) gauge invariance to describe electromagnetic interactions [33]. Furthermore, the formulation of QFT taking Lorentz invariance from special relativity and probabilistic interpretation from quantum theory is presented with an immediate issue—Lorentz transformations mix positive and negative norm states due to the (+, −, −, −) signature of the metric and therefore probabilities are not bounded. Gauge invariance plays a special role in removing negative norm states (for a detailed example for a spin-1 field, see [34]) which are responsible for the rapid rise of the scattering amplitudes at high energies. These states are unphysical and named as ghost states. The current formulation of the Standard Model describes three of the four fundamental interactions under a unified framework employing the notion of local gauge invariance. Quantum chromodynamics [35–37] describes the strong interactions with a coloured gauge group SU (3)c and in the Weinberg-Salam theory, the weak and electromagnetic interactions are unified [38–40] within the electroweak gauge group SU (2) L ⊗ U (1)Y . Terms in the Lagrangian are constructed from fields in Table. 1.1 so that they are a gauge singlet under: G S M = SU (3)C ⊗ SU (2) L ⊗ U (1)Y .

(1.8)

Table 1.1 Group structure of the SM particles under G S M (cf. Eq. 1.8) and the Lorentz group Fields SU (3)c SU (2) L U (1)Y S L(2, C) 1 1 q Li 3 2 ,0 6 2 1 2 0, u iR 3 1 3 21 0, d Ri 3 1 − 13 1 2 1 i L 1 2 −2 ,0 2 1 eiR 1 1 −1 0, 2 1 1 G aμ 8 1 0 , 21 21 Wμi 1 3 0 , 21 21 Bμ 1 1 0 2, 2 1 0) H 1 2 (0, 2

6

1 Introduction—Realisation of the EW Symmetry in the SM

Invariance under this gauge group demands the matter fields to transform as:    σi i λa a ψ(x) → exp i g1 Y θ1 (x) + g2 θ2 (x) + g3 θ3 (x) ψ(x), 2 2

(1.9)

where g1 , g2 and g3 are the coupling constants of U (1)Y , SU (2) L and SU (3)c respectively. To build terms consistent with Poincaré and gauge group symmetry, we consider as an example, the representation of the kinetic term of a left-handed quark doublet under (1.8): i

Q L γ μ Dμ Q L

∼

(3, 2, −1/6) × (3, 2, 1/6) = (1, 1, 0) + · · ·

Here the covariant derivatives:   σi i λa a Dμ = ∂μ − ig1 Y Bμ − ig2 Wμ − ig3 G μ , 2 2

(1.10)

(1.11)

are responsible for keeping the terms (involving ∂μ ) invariant with (1.8) being local symmetry. This necessarily introduces the gauge fields B, W i and G a for the respective groups and their transformations: G aμ (x) → G aμ (x) + ∂μ θ3a (x) + g3 f abc θ3b (x)G cμ (x), j

Wμi (x) → Wμi (x) + ∂μ θ2i (x) + g2 i jk θ2 (x)Wμk (x),

(1.12)

Bμ (x) → Bμ (x) + ∂μ θ1 (x). Based on this principle, the kinetic terms of fermions and bosons can be constructed:    1 I I μν μ − Fμν F − ψiγ Dμ ψ , (1.13) L⊃ 4 a i Fμ ∈{G μ ,Wμ ,Bμ } ψ∈{Q iL ,iL ,u iR ,d Ri ,iR }

with the field strength tensors defined as: G aμν = ∂μ G aν − ∂ν G aμ − g3 f abc G bμ G cν , i Wμν = ∂μ Wνi − ∂ν Wμi − g2 i jk Wμj Wνk ,

(1.14)

Bμν = ∂μ Bν − ∂ν Bμ . One quickly runs into trouble when constructing mass terms (cf. Table 1.2) for the fermions and vector gauge bosons of the form: Lmass ⊃ −m 2 V μ Vμ − mψ L ψ R + h.c.,

(1.15)

1.1 Renormalisability, Unitarity, Gauge Invariances and all that Table 1.2 Masses of the SM particles. Source: [41] Particle Mass Particle νe νμ ντ u d

< 225 eV < 0.19 MeV < 18.2 MeV 2.3+0.7 −0.5 MeV 4.8+0.5 −0.3 MeV

e μ τ s t

c W± Z0

1.275 ± 0.025 GeV 80.385 ± 0.015 GeV 91.1876 ± 0.0021 GeV < 1 × 10−18 eV 0

b h

γ g

7

Mass 0.5110 MeV 105.7 MeV 1177 MeV 95 ± 5 GeV 173.21 ± 0.51 ± 0.71 GeV 4.66 ± 0.03 GeV (1S) 125.09 ± 0.25 GeV

which are SU (2) L ⊗ U (1)Y invariant (even in the global sense). Furthermore, the scattering of the weak-bosons once again jeopardises perturbative unitarity. To introduce new physics without spoiling gauge invariance or renormalisability, one makes the mass spurious by promoting it to the Higgs field which transforms as (1, 2, 21 ). It suffices to then consider the representation of the following terms under global G S M : −y  L H e R + h.c.

∼

−yd Q L H d R + h.c.   −yu Q L iσ 2 H ∗ u R + h.c.  

∼

Dμ H † D μ H

∼

∼

:= H˜

   1, 2, − 21 ⊗ 1, 2, 21 ⊗ (1, 1, 1)       3, 2, − 61 ⊗ 1, 2, 21 ⊗ 3, 1, − 13       3, 2, − 61 ⊗ 1, 2, − 21 ⊗ 3, 1, 23 



   1, 2, 21 ⊗ 1, 2, − 21 ⊗ (1, 3, 0)⊗2

= (1, 1, 0) + · · · = (1, 1, 0) + · · · = (1, 1, 0) + · · ·

= (1, 1, 0) + · · ·

(1.16) To generate the mass term, the idea of spontaneous symmetry breaking is required for the Higgs to develop a vacuum expectation value, which we introduce in the next section.

1.2 Spontaneous Symmetry Breaking and the Higgs Mechanism In QFT, two particle states related by a symmetry generated by a Nöther charge, Qˆ are degenerate if and only if the vacuum of the theory is invariant under the symmetry ˆ (i.e. Q|0 = 0). Spontaneous symmetry breaking occurs when the physical vacuum of the theory is not invariant under the symmetry of the action (Lagrangian). In

8

1 Introduction—Realisation of the EW Symmetry in the SM

order to grant masses to SM particles through spontaneous symmetry breaking at the electroweak scale, one has to evade the theorem due to Goldstone. Theorem 1.2.1 (Goldstone’s Theorem [42, 43]) If a theory possesses a continuous symmetry, then either the vacuum state is invariant under such symmetry, otherwise there will be a corresponding spinless particles of zero mass for each broken generator of the vacuum. This is illustrated with a theory which is invariant under some symmetry group action: ⎛ ⎞ φ1 ⎜ .. ⎟ a δφ = ia T φ, (1.17) φ = ⎝ . ⎠, φn ,

and possess a potential V (φ) obtains a minimum at φ = φ such that: ∂V (φ) = 0, ∂φi

∂2 V (φ) = m i2j . ∂φi ∂φ j

(1.18)

where m i2j is the mass matrix for the scalar fields. The generators of the symmetry G can be divided into those that annihilate the vacuum, Y i , and those that do not, X aˆ , such that: (T a ) = (Y i , X aˆ ),

Y i φ = 0,

X aˆ φ = 0,

(1.19)

with Y i generating a subgroup H ⊂ G. When the symmetry the Lagrangian is spontaneously broken from G to H, the Lagrangian (potential) is still invariant under the latter subgroup: 0 = V (φ + δφ) − V (φ) = i

∂V a a  (T )i j φ j . ∂φi

(1.20)

Differentiating this with respect to φk at φ = φ: 0 = ia (T a )i j φ j

∂2 ∂ 2 a V (φ) + ia (T a )ik V (φ) = im ik  (T a )i j φ j . (1.21) ∂φi ∂φk ∂φi

The interpretation is that for each broken generator in the coset space G/H, there is an associated zero eigenmass. We refer the reader to the original papers and [44] for further details. Let us return briefly to Fermi’s effective theory, where unitarity violation may be cured by replacing the d = 6 operator by a d = 4 (and hence renormalisable) one via the introduction of intermediate vector bosons [40, 45–50]:

1.2 Spontaneous Symmetry Breaking and the Higgs Mechanism

GF LFermi ⊃ √ Jμ J μ 2

→

g2 L E W ⊃ − √ Wμ+ J −μ + h.c., 2 2

9

(1.22)

√ with G F = ( 2v 2 )−1 being the coupling constant. Although the doublet structure of the current3 :  i   i   σi u ν , (1.23) ψ∈ , Jμ = ψγμ ψ, di L i L 2 seems to naturally fit a non-Abelian gauge theory [51, 52] with a SU (2) L symmetry4 , barely any reference (except in [48, 50]) were made regarding the gauge structure. This is due to the dilemma that spontaneous breaking of global gauge symmetry supposedly predicts massless modes which can’t be identified with the massive gauge bosons5 . The solution to evading Goldstone’s theorem is to make the broken symmetry local [57–62]. This way, the would-be Goldstone boson, endows the massless gauge boson with mass and are removed from the massless spectrum. As we see in the next part, this may actually be achieved without the addition of a Higgs boson.

1.2.1 Non-linear Realisation Since all the mass terms are already invariant under SU (3)c ⊗ U (1) Q , one only needs to gauge the coset group SU (2) L ⊗ U (1)Y /U (1) Q . This is naturally described in the formalism due to Callan, Coleman, Wess and Zumino (CCWZ) [5, 6]. We review this framework in order to provide a parameterisation for the Goldstone bosons. Suppose the scalar field of the theory maps φ : R(1,3) → M and that it possesses a symmetry G. If the subgroup H < G forms the stabiliser which leaves the vacuum φ invariant: hφ = φ,

∀ h ∈ H,

(1.24)

and that the vacuum of the theory is connected, then G generates the flat directions of the potential forming the vacuum manifold: Vac(M) := Gφ ∼ = G/H,

(1.25)

where the isomorphism follows from the orbit-stabiliser theorem. We will assume that the generators in (1.19) are orthonormal with respect to the Cartan-Killing inner product: 3 At

the time, hadronic currents were described by

   i p u instead of , before there was n L di L

convincing evidence of quarks. the subscript ‘L’ is due to the left-handed structure in accord with parity violation [53–56]. 5 The bosons are required to be massive in order to explain the short-ranged behaviour of the weak interaction. 4 Here

10

1 Introduction—Realisation of the EW Symmetry in the SM

Tr(T a T b ) = δ ab , T a , T b ∈ {X aˆ , Y b }.

(1.26)

The setting is then on an analytic manifold M with a Lie-group G-action: G : G × M −→ M, (g, φ(x)) −→ Tg φ(x).

(1.27) (1.28)

The coordinates of the left coset manifold G/H6 are represented by the masses Goldstone modes π(x), which are identified as the quantum excitations along the flat directions of the vacuum. Since the generators {X aˆ , Y b } are orthogonal, multiplication of coset representatives in G/H by g ∈ G may be uniquely decomposed as: ge−iπ

aˆ

X aˆ

= e−iπ

aˆ

X aˆ

h.

(1.29)

The idea is to then use (1.24) and parameterise fields on M as: φ(x) = U φ = e−iπ where according to (1.29):

aˆ

X aˆ

φ.

g

→ gU h −1 (g, U ). U −

(1.30)

(1.31)

It is then always possible to use π(x) as the coordinates of the vacuum submanifold with origin at φ so that the action of h ∈ H on the remaining coordinates ψ(x) is linear [5, 63]. G then is non-linearly realised on the coordinate pair: g

(π, ψ) −−−→ (π  , ρ(h(g, U ))ψ).

(1.32)

In fact, the non-linear shift symmetry of the Goldstone fields from (1.31) forbids non-derivative terms. The simplest term that is G-invariant is U † U = I. The next object of interest is then the Maurer-Cartan one form [6]: ωμ = U † ∂μ U = idμaˆ X aˆ + i E μi Y i .

(1.33)

It follows that the G-action:





U † ∂μ U −→ hU −1 g −1 ∂μ gU h −1 = h U −1 ∂μ U h −1 + h∂μ h −1 , (1.34) implies that dμ := dμaˆ X aˆ transform linearly and E μ := E μi Y i non-linearly, taking the role of a gauge field: dμ −→ hdμ h −1 , (1.35) E μ −→ h E μ h −1 + h∂μ h −1 . 6 G /H

:= {[l], l ∈ G } where [l] := {g ∈ G | l −1 g ∈ H} so that g = l(l −1 g) ∈ l H.

1.2 Spontaneous Symmetry Breaking and the Higgs Mechanism

11

Particularly, the dynamics of the Goldstone bosons is encoded at the two derivatives level √ by the Lagrangian (canonically normalising the dimensionless π fields via π → 2π/ f ): L(2) =

f π2 μ Tr d dμ = ∂μ π aˆ ∂ μ π aˆ + · · · 4

(1.36)

Interaction terms between the Goldstone modes and the ψ fields will then give the effective low energy dynamics. The Lagrangian should be independent of whether it is constructed out of {π aˆ , ψ, ∂μ π, ∂μ ψ} or {U, ψ, dμ , ∇μ ψ}, where the covariant derivative of ψ is constructed using the connection E μ : Dμ ψ := ∂μ ψ + E μ ψ.

(1.37)

Let us apply this to the electroweak theory so that the Goldstone modes parameterising the coset space G/H = SU (2) L ⊗ U (1)Y /U (1) Q ∼ = SU (2) are embedded into a matrix:  := e− v π i

a

(δ a3 I−σ a )

,

Dμ  := ∂μ  + ig2

σa a σ3 Wμ  + ig1  Bμ . (1.38) 2 2

Since the unbroken generator is Q = 21 (σ 3 + Y ), (1.31) gives the transformation: (x) −→ e−iβ e−iα

a σa 2



(x)e 2 β (I−σ ) . 1

3

(1.39)

The minimal Lagrangian describing the low energy SU (2) L ⊗ U (1)Y sector at two derivatives is then given by: LEWχ

    u j

v v2 λ u i ⊃ Tr Dμ  D μ  † − √ u iL d L  idj Rj + h.c., 4 λi j d R 2

(1.40)

where summation over the flavour indices i and j is assumed. Expanding the covariant derivative in powers of Goldstone field results in: ⎡

  ⎤    ∂μ π a ∂μ π 3 g g g 2 2 1 a a 3 W σ + √ W + Bμ σ 3 ⎦ + · · · − − Dμ  = i ⎣ √ 2 μ 2 μ 2 2v 2v a∈{1,2}

(1.41)

The kinetic term from (1.40) may be written in terms of the expansion above as:  2 2     g2 1 g2 v2  v Z μ  , Tr Dμ  † D μ  = ∂μ π + − √ vWμ+  + ∂μ π 3 − √ 2 2 2 2 cos θw

(1.42)

where one recovers the gauge boson mass terms when the theory is expanded around the vacuum  = I:

12

1 Introduction—Realisation of the EW Symmetry in the SM

Lmass ⊃

g22 v 2 + μ− g22 + g12 2 μ Wμ W + v Z Zμ. 4 8

(1.43)

Such a chiral effective Lagrangian remains consistent without the need of a Higgs up to a scale of  ∼ 1 − 3 TeV. In the absence of new physics, the theory becomes strongly coupled [64–72]. This can be argued from unitarity considerations where the Goldstone equivalence theorem provides a key relation between the longitudinal W boson scattering (i.e. W L W L → W L W L ) and that of the corresponding Goldstone bosons π + π − → π + π − : Theorem 1.2.2 (Goldstone equivalence theorem [26, 73–76]7 ) In a renormalisable gauge theory which is spontaneously broken, the scattering amplitude of longitudinally polarised gauge bosons is equivalent the amplitude where the external gauge boson legs are replaced by the corresponding Goldstone bosons: M (W L ( p1 ), ..., W L ( pn ) + X → W L ( p1 ), ..., W L ( pm ) + X )    mW = (−i)m+n M (π( p1 ), ..., π( pn ) + X → π( p1 ), ..., π( pm ) + Y ) 1 + O √ . s

(1.44) This can be pictorially represented as follows:

 =

×

 1+O

mW √ s

 .

(1.45) To apply this, the first term of (1.40) has to be written in terms of the would-be Goldstone:  "

2 v2  1 1 ! Tr ∂μ ∂ μ  = ∂μ π + 2 (π a ∂μ π a )2 − π a π a ∂μ π b ∂ μ π b + O(π 6 ), 4 2 6v

(1.46)

which gives the amplitude for π + π − → π + π − scattering of the form:

A(π + π − → π + π − )=

1 (s + t). v2 (1.47)

7 For

proof in chiral Lagrangians, see [77–80].

1.2 Spontaneous Symmetry Breaking and the Higgs Mechanism

13

The unitarity bound prescription (cf. App. A.1) asserts that each partial wave: 1 al := 32π



1 −1

d cos θ A(s, θ)Pl (cos θ),

(1.48)

has to satisfy Re(al ) ≤ 1/2, with Pl being the l-th Legendre polynomial. The s-wave amplitude: a0 (W L+ W L− → W L+ W L− ) 

1 s , 32π v 2

(1.49)

then gives a scale  ∼ 4πv at which perturbative unitarity is violated.

1.2.2 ST U Precision Parameters Before we discuss a UV completion of the theory by addition of a scalar particle, we turn to two operators which we have neglected in (1.40): LEWχ ⊃

     a σ3 v2 σ3 a σ aT Tr  † Dμ  + a S Tr  † Wμν  B μν , 2 2 2 2

(1.50)

where the coefficients a S and aT contribute to the S and T parameters [81, 82]:  2 3 αW (m Z )  , ln 16π 2 cos2 θW m 2Z  2 g22 αW (m Z )  ˆ .  S = a S () + ln 12π 4 sin2 θW m 2Z

Tˆ = aT () −

(1.51)

The logarithmic running of these precision parameters are due to the one-loop exchange of the Goldstone bosons [83] as shown in Fig. 1.1. These electroweak precision variables are strongly constrained by LEP (Large Electron-Positron Collider) [41]. To introduce these we write the gauge sector of the Lagrangian terms of the vacuum polarisations: 1

μν

1

μν

μν

μν

L ⊃ − Wμ3 33 (q 2 )Wν3 − Bμ 00 (q 2 ) − Wμ 30 (q 2 )Bν − Wμ+ W W (q 2 )Wν− , 2 2

where the correlators are defined via:   μ ν q q μν 2 μν i j (q 2 ). −η i j (q ) := q2

(1.52)

(1.53)

14

1 Introduction—Realisation of the EW Symmetry in the SM

Fig. 1.1 Quantum corrections the self-energies of the gauge and would-be Goldstone modes

π 2 (π 1 )

π2 W3

B

π 1 (π 2 )

π1

π 1 (π 2 ) B

The parameters in (1.51) are then related to the Peskin-Takeuchi S, T and U parameters [84] via: Sˆ =

g2   (0), g1 30 33 (0) − W W (0) , = m 2W U = 33 (0) − W W (0).

α S = 4 sin2 θW

Tˆ =

αT α Uˆ = − 4 sin2 θW

(1.54)

As deviations in such parameters have been constrained to be  10−3 , the appearance of the operators from (1.50) can only be incorporated in the phenomenological Lagrangian with very small a S and aT . The smallness of the T parameter can be explained by the accidental custodial symmetry. The approximate global SU (2) L ⊗ SU (2) R symmetry assumed by the chiral Lagrangian is broken by the vacuum  = I into the diagonal subgroup SU (2)c . This custodial symmetry protects the tree level relation: ρ :=

m 2W = 1, m 2Z cos2 θW

(1.55)

and is explicitly broken by operators such as that in (1.50), giving corrections to (1.55) proportional to g1 and λiuj − λidj . In the heavy top limit, the large mass difference originates from a dimensionless constant that becomes large, giving large one loop contributions to ρ = 1 + Tˆ . This is an example where the Appelquist-Carazzone theorem does not apply and heavy physics is not decoupled.

1.2.3 Higgs as Singlet Addition from Unitarity Considerations The simplest UV completion of the chiral model is provided by a new singlet scalar resonance h under the EW gauge group, which moderates the high energy behaviour of the scattering amplitude. In order to keep corrections to Tˆ = ρ − 1 small, this resonance has to also be a singlet under the custodial symmetry. The Lagrangian up to two derivatives then becomes:

1.2 Spontaneous Symmetry Breaking and the Higgs Mechanism (2)

L

15

 

h2 h3 h v2 μ † 1 + 2a + b 2 + b3 3 + · · · ⊃ ∂μ h∂ h − V (h) + Tr Dμ  D  4 v v v   u i    2  h h v λi j u R i + h.c., u iL d L  1 + c + c2 2 + · · · −√ λidj d Ri v v 2 i, j∈{1,2,3} μ

(1.56) where the potential is given by : 1 V (h) = m 2h + d3 2



m 2h 2v



 h + d4 3

m 2h 8v 2

 h4 + · · ·

(1.57)

The following part will show that the SM is a specific configuration of this model which requires a = b = c = d3 = d4 = 1 and b3 = c2 = 0 in order to unitarise the scattering amplitudes. Let us first re-analyse the scattering amplitude, following [85], for π + π − → π + π − (cf. Fig. 1.2): ⎡

⎤

1 ⎢ A(π + π − → π + π − ) = 2 ⎣−(s 2 + 4st + t 2 ) +

  v Z /γ exchanges

s+t = 2 (1 − a 2 ) + O v



m 2h E2

(s 2 + 4st + t 2 )

 

+s − a 2

direct W W W W coupling

s2 s − m 2h

⎥ + (s ↔ t)⎦



.

(1.58) Particularly, the cancellation (Fig. 1.3) of the O(E 4 ) term is due to gauge invariance, while that of the O(E 2 ) terms is due to the specific Higgs couplings to the electroweak gauge bosons as tuned by a. One has to also consider the amplitudes for π + π − → ψψ and π + π − → hh: where the scattering amplitudes are given by: √  2 mh mψ s , A(π π → ψψ) = (1 − ac) + O v2 E2  2 mh s A(π + π − → hh) = 2 (b − a 2 ) + O . v E2 + −

π−

π−

π−

π−

π− h

h π+

π+

π+

π−

π+

π+

Fig. 1.2 Feynman diagrams for the scattering of the Goldstone modes

π+

16

(a) π−

1 Introduction—Realisation of the EW Symmetry in the SM

ψ

(b)

ψ

π−

(c) π−

(d) h

π− π

h π+

ψ

π+

h

ψ

π+

h

π+

h

Fig. 1.3 Feynman diagrams for the scattering of the Goldstone modes via π + π − → ψψ in (a) and (b), and via π + π − → hh in (c) and (d)

The unique point in theory space that unitarises all the considered amplitudes is then given by8 : ⎫ a2 = 1 ⎪ ⎬ b = a2 ⎪ ⎭ ac = 1

=⇒

a = b = c = 1.

(1.59)

The SM Higgs doublet may be identified as with the point (1.59) in the non-linear realised theory space as9 :   1 0 H (x) := √  . v + h(x) 2

(1.60)

We make clear the difference between the scenarios10 : • Non-linear realisation Generically, this occurs whenever a theory with symmetry G is broken to one with H. The relevant case here is when the Goldstone parameterises the coset group SU (2) L ⊗ U (1)Y /U (1) Q , which corresponds to the symmetry breaking of the SM EW gauge group. Imposing perturbative unitarity, the minimal addition required is a Higgs scalar, h which is a singlet under the full group G S M . This way the interaction of the Higgs is not necessarily correlated with that of the longitudinal components of the W ± and Z bosons. • Linear realisation In a sense, this is just a specific non-linear realised model. The SM is one such example where the singlet h and non-linearly realised Goldstones

are fixed into a linear Higgs complex doublet structure H , transforming as 1, 2, 21 under (1.8). The S-matrices of the theory should be unchanged by treating (1.60) We note that similar considerations for non-minimal scenarios where a SU (2) L multiplet is added instead of a singlet have been carried out in e.g. [86, 87]. 9 This identification can only be made if v + h > 0. 10 We note the recent series of works [88–90] are suggesting that the SM represents a flat manifold for the singlet scalar, and that deviations from the SM may be parameterised by introducing curvature to the manifold. 8

1.2 Spontaneous Symmetry Breaking and the Higgs Mechanism

17

as a coordinate transformation [91–93], should this be used to identify the SM Higgs multiplet within the non-linearly realised framework. The recent focus was to advocate the use of effective field theory to look for new physics. The common approach has been to view the SM as part of an EFT with the EW gauge group linearly realised: (4) (4) (4) 4 L = c0 U V + LHiggs + LGauge + LYukawa +

1 1 L(5) + 2 L(6) + · · · U V U V

(1.61)

There is one operator in L5 [94]. In L6 there are 59 (+ h.c.) operators that conserve baryon number [95, 96] and four that does not [95, 97]. All twenty operators in L7 violate lepton number [98]. L8 was recently classified, and contains a huge 535 (+ h.c.) operators [99] for one flavour, 46 of which violate baryon number. Such a linear EFT relies on a hierarchy between the electroweak and new physics scale and so the interest is in obtaining global constraints on leading contributions coming from the dimension-six Wilson coefficients, c6 . However, this approach was met with considerable difficulties (cf. e.g. [100–103]). On one hand, when new physics enters at too high a scale, effects from the non-renormalisable operators become experimentally unobservable. On the other hand a substantial theoretical error on c6 may be induced by dimension-eight operators when one considers mille-level precision on electroweak data with cut-off scale   1 TeV [104]. Nevertheless, the use of such approach has been shown to well describe extensions to the gauge-Higgs sector, provided that it remains weakly interacting (see e.g. [105]). The generation of non-Abelian gauge boson masses by spontaneous symmetry breaking in the SM is unique from renormalisability11 [110–112] and unitarity [26, 74–76, 113, 114] considerations. Furthermore, the series of papers [115–117] have demonstrated non-powercounting renormalisability as a property of non-linearly realised models12,13 . In light of these discussions, it is important to consider the realisation of the electroweak gauge group, especially when sizable deviations from the linearly realised structure of the SM are still allowed by the LHC data [130, 131]. Without exact knowledge of the exact symmetry breaking sector, one should therefore follow the generic non-linear prescription where the h(125) resonance is treated as a 11 In fact, a stronger statement that renormalisable models must be equivalent to an spontaneously broken quantum theory was shown in [26]. Renormalisability of spontaneously broken Yang-Mill theories has been studied in [106–109]. 12 This has subsequently revived attention to the Stückelberg mechanism [118] (for a recent review see [119]) which is able give gauge invariant mass terms without the Higgs in the perturbative spectrum but by introduction of a flat connection on the coordinate manifold of the Goldstone modes (cf. [120] for a more pedagogical discussion). This was applied to SU (2) L ⊗ U (1)Y theory [121–123] but was found to spoil the custodial relation (1.55). In light of the Higgs discovery, it is important to check if there is indeed a Stückelberg component in the mass. To this end, [124, 125] have found that four scalar resonances will be required to keep the nice property of weak power counting renormalisability (cf. e.g. [126]) and that tree-level unitarity will be violated by the slightest contribution from the Stückelberg sector. 13 See also [127–129] for renormalisation of non-linearly realised Higgs models.

18

1 Introduction—Realisation of the EW Symmetry in the SM

custodial and SM singlet coupled to the electroweak chiral Lagrangian (cf. Eq. 1.56). The community seems be arriving on a consensus to adopt such approach [65, 67, 69–71, 132–137]. In such cases, CP-violating Yukawa couplings may result from complex λiu,d j . Putting constraints on such couplings in the top-Higgs sector is then the subject of Chap. 3. Furthermore, the trilinear Higgs coupling is no longer proportional to the quartic coupling as in the SM, instead becoming unconstrained except via naïve dimensional analysis arguments. The implications of these new physics effects on electroweak baryogenesis will be studied in Chap. 4.

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Chapter 2

Spin Determination of the LHC Higgs-Like Resonance

The Higgs discovery was initially made in the diboson modes h → Z Z ∗ , W W ∗ , γ γ . The spin-parity assignment of the newly discovered boson h constitutes the first step to establish whether this particle is the SM Higgs responsible for electroweak symmetry breaking, or merely an impostor. We first discuss in Sect. 2.1 that the observation of the on-shell diphoton decay rules out the J = 1 assignment due to the well known Landau-Yang theorem. The most immediate cases alternate to the SM Higgs is that of a pseudoscalar (J P = 0− ) and spin-2 (J = 2). It was argued in [1, 2] that the alternate hypotheses are disfavoured by the analysis of the observed production and decay rates. In particular, the measured h → gg and h → γ γ rates are consistent with that of a SM Higgs:  S M (h → gg) ≈ 37  S M (h → γ γ ).

(2.1)

This is in tension with many spin-2 models with compactified extra-dimensions which predicts a universal coupling to photons and gluons cg = cγ where a ratio of 8 should be observed between respective rates. Furthermore, it was noted in [1, 3] that universal couplings h μν D μ  † D ν  preserves gauge invariance and thus custodial symmetry. The approximate relation cW ≈ c Z was found to be consistent with the available data. On the other hand, the methods adopted in the experimental analysis relies on studying the kinematic distributions (cf. [1–16] and references therein) which are expected to be highly anisotropic for higher spins. The reason why higher spins (J ≥ 3) are not considered is because (i) of angular momentum conservation arguments, and (ii) that in these cases the number of independent helicity amplitudes for the Z Z ∗ decay does not exceed that of six as in J = 2 and can be distinguished based on the energy dependence [17, 18]. The framework can be made model independent by writing down the most general amplitude [17, 19, 20] or Lagrangian compatible with Lorentz and gauge symmetry. In particular, the use of effective field theory [8, 21, 22] has the following advantages: (i) it allows a hierarchy of new physics effects © Springer International Publishing AG 2017 J.T.S. Yue, Higgs Properties at the LHC, Springer Theses, DOI 10.1007/978-3-319-63402-9_2

25

26

2 Spin Determination of the LHC Higgs-Like Resonance

to be ordered with respect to the energy scale, and (ii) provides a framework to also study the associated anomalous couplings. At the time of writing [23], ATLAS had placed 99.9% C.L. exclusions using the W W , Z Z and γ γ channel regardless of the relative contribution of gg or qq production [24]. CMS reached a similar conclusion at 99.4% C.L but using only the W W and Z Z modes with the gg-initiated processes [25]. Whilst these constitute strong evidence against a spin-2 scenario under the assumption of minimal graviton couplings, they are not conclusive for a spin-2 impostor with generic couplings. As explained in Sect. 1.2, a strong motivation for the existence of the Higgs boson is that it unitarises the high energy behaviour of vector boson scattering (see Appendix A.1 for a discussion of unitarity). The aim of our Letter (cf. Ref. [23]) is then to demonstrate that a J = 2 impostor mimicking the decay rates of the 125 GeV signal would jeopardise this feature. Rather than reproducing the entirety of the Letter, its motivation and calculations will be detailed in this chapter. Firstly the LandauYang theorem is presented in Sect. 2.1. Subsequently, the theoretical difficulty of constructing an interacting spin-2 theory is addressed in Sect. 2.2, which explains why the use of an EFT is necessary. In Sect. 2.3, we calculate the cutoff-scale for a theory with a spin-2 Higgs-impostor, h. This is taken to be the scale of unitarity violation for the longitudinal scattering of h Z → h Z , which we found to be significantly lower than the LHC reach. We conclude with remarks on the Letter in Sect. 2.5.

2.1 Excluding J = 1 In this section we give a discussion on the Landau-Yang theorem, which is stated as follows: Theorem 2.1.1 (Landau-Yang [26, 27]) A spin-1 (J = 1) state cannot decay into two identical massless vectors. We inspect this briefly with an Abelian case where the vectors are photons. Interested readers are referred to [28] for the generalisation to non-Abelian cases. We first denote the amplitude h(J = 1) → γ γ as: M(1, 2) := γ (k1 , ε1 )γ (k2 , ε2 )|h( p, ε),

(2.2)

where k1,2 and ε1,2 are the four momenta and polarisation of the photons respectively. Transversality then requires that ε · p = 0, whilst conservation of momenta requires that p = k1 + k2 . We define q := k1 − k2 so that p 2 = −q 2 = m 2h and p · q = 0. Gauge invariance is preserved when the vector polarisations are boosted by currentconserving objects: εi = εi −

εi · k j k j , (i, j) ∈ {(1, 2), (2, 1)}, ki · k j

(2.3)

2.1 Excluding J = 1

27

  such that p · ε1,2 = q · ε1,2 = 0 is satisfied. The most general Lorentz invariant amplitude1 must be linear in all the polarisation-vectors, taking the form:

  M(1, 2) := F1 ( p 2 ) ε1 · ε2 (ε · q) + F2 ( p 2 ) (ε · q) μνρσ ε1μ ε2ν p ρ q σ μ

+ F3 ( p 2 )μνρσ ε1 ε2ν ερ p σ .

(2.4)

The Landau-Yang theorem then follows as the amplitude violates Bose symmetry, i.e. M(1, 2) = M(2, 1), under the interchange of photons (ε1 ↔ ε2 and q → −q). This is true whenever the functions Fi ( p 2 ) are not all zero [29]. Although it was pointed out that the theorem may be evaded2 in the J = 1 hypothesis because of on-shell assumptions [32–34], this results in the lack of resonance structure which contradicts with the data [8].

2.2 Massive Spin-2 The construction of a consistent massive interacting spin-2 theory is met with various difficulties (see discussion in e.g. [35–40]), with the most prominent being the existence of ghost states [40]. Construction of a free Lagrangian for massive higher spins is done in [41, 42] and a unique Fierz-Pauli structure3 applies to the spin-2 case [44–46]:   1 μν h ρσ − m 2h h μν h μν − h 2 , (2.5) L J =2 ⊃ h μν Eρσ 2 μν is: where h := h μμ and the definition of the linearised Einstein operator Eρσ μν Eρσ h ρσ :=

1 1 h μν − ∂α ∂(μ h αν) + ∂μ ∂ν h − ημν h. 2 2

(2.6)

Five constraint equations are then required to reduce the ten degrees of freedom in the symmetric h μν field, and can be obtained by acting with ∂μ and 21 ημν + m12 ∂μ ∂ν h on the equation of motion:   μν − m 2h h μν − hημν = 0. Eρσ

(2.7)

One subsequently arrives at the gauge conditions: the photon energy E in the rest frame of h(J = 1), the momentum q assumes the form (0, 0, 0, 2E ). Furthermore, with ε1,2 being the only possibilities for the transverse polarisations of μ the massless photon, the term μνρσ ε1 ε2ν ερ p σ must be proportional to ε0 ∼ |p|/m h = 0 in this rest frame and is hence neglected in (2.4). 2 This is also possible if the functions F ( p 2 ) in (2.4) are anti-symmetric and hence recover Bose i symmetry (cf. e.g. [30, 31] for a discussion on Bose symmetry violation). 3 It was however shown in [43], that there can be a consistent theory that violates this structure. 1 Given

28

2 Spin Determination of the LHC Higgs-Like Resonance

∂μ h μν − ∂ν h = 0, h = 0.

(2.8)

From this, it is seen that the coefficients (in particular the mass term) are tuned to avoid ghost terms. The gauge invariance condition δh μν = ∂μ ξν + ∂ν ξμ can be restored in the m = 0 case by introducing extra degrees of freedom, Aμ and χ , following the same pattern (so h μν → h μν + ∂(μ Aν) and Aμ → Aμ + ∂μ χ ): h μν → h μν +

1 1 ∂(μ Aν) + m 3m 2

  1 ∂μ ∂ν χ + m 2 ημν χ . 2

(2.9)

The mass term in (2.5) then becomes:   1 1 1 Fμν F μν + χ χ − m 2h h μν h μν − h 2 8 12 2   mh 1 1 + m 2h χ 2 + χ h + m h h∂μ Aμ − h μν ∂μ Aν + χ ∂μ Aμ , 6 2 2

L⊃−

(2.10)

where the helicity states—h μν for helicity-two, Aμ for helicity-one and χ for helicityzero, of the massive spin two states are manifest. Technically, the gauge symmetry is augmented by the Stückelberg sector [40]: 1 h μν → h μν + ∂(μ ξν) + ημν m, 2 Aμ → ∂μ  − mξ, χ → χ − 3m,

(2.11)

where the shift in the metric is to decouple the kinematic mixing. The Stückelberg formalism [47] then emphasises that gauge invariance are not real in the sense that they are just redundancies in the description of a theory.

2.2.1 Couplings to Matter There are several issues in constructing interaction terms for spin-2 states (for more pedagogical discussions, see [48, 49]). This is mainly attributed to the lack of currents invariant under gauge symmetry associated with such higher spin states (cf. e.g.[50]). We study the coupling of a linear theory to a source Tμν of the form: L = L J =2 +

κ h μν T μν . 2

(2.12)

Taking the motivation from the Einstein-Hilbert action (augmented by the SM Lagrangian):

2.2 Massive Spin-2

29

 S=

√ d 4 x −g (R + L S M ) ,

(2.13)

√ one arrives at (2.5) by expanding g R with gμν = ημν + κ2 h μν . Then (2.12) can be viewed as an expansion of (2.13): S[gμν ] = S[ημν ] +

κ 2



  d 4 x h μν Tμν + O κ 2 .

(2.14)

with the source identified as follows:  2 δS  Tμν := − √ , −g δgμν gμν =ημν

(2.15)

The divergencelessness of the current is crucial in keeping diffeomorphism4 . To demonstrate this, take as an example the source term for a massless scalar in (2.5): 1 (0) = ∂μ φ∂ν φ − ημν ∂α φ∂ α φ, Tμν 2

(2.16)

where the conservation of the energy momentum tensor and gauge invariance is ensured by the on-shell condition φ = 0. However, due to the additional term in (2.12), the Klein-Gordon equation is modified to:    1  ν μ   μ ν φ = κ ∂ h μν ∂ φ − ∂μ h ν ∂ φ . 2

(2.17)

(0) broken at order The theory is then inconsistent, with the conservation of Tμν O(κ):     1   (0) ∂ μ Tμν = κ∂ν φ ∂ α h αβ ∂ β φ − ∂μ h αα ∂ μ φ . (2.18) 2

One needs to introduce couplings of order O(h 2 ), so that the (non-linear) self-energy of the spin-2 will be included in Tμν , equivalent to a second-order expansion in (2.14). Iterative applications of these corrections should recover (2.13) but with the diffeomorphism promoted to the fully non-linearly realised gauge symmetry (coordinate covariance) [37, 51–56]. To make the dynamics of different helicity components of the massive spin-2 more obvious, one uses again the Stückelberg extension (2.9) to generate the following terms in the Lagrangian: L⊃

4 We

κ κ κ κ h μν T μν + χ T μμ − Aμ ∂ν T μν + 2 χ ∂μ ∂ν T μν . 2 2 mh mh

(2.19)

note that in the case of a massless and sourceless Einstein-Hilbert action, this follows as a natural consequence of the linearised Bianchi identity.

30

2 Spin Determination of the LHC Higgs-Like Resonance

We see that if Tμν is not divergenceless, the scalar and vector modes will be strongly coupled to Tμν in the m h → 0 limit5 . In addition, the helicity-zero component never decouples due to the second term, and is the origin of what is known as the vDVZ discontinuity [57, 58]. This was suggested not to be a problem per se (see e.g. [59]), since the non-linearities of the theory should be important at scales  1/5 V ∼ m 4h 2 κ [60], with  being the intrinsic cutoff of the linear theory. The real issue is that when these non-linear effects are included by changing the Pauli-Fierz structure, Boulware-Deser ghost [61] propagate as a sixth degree-of-freedom. The construction of a ghost-free theory is contrived [62–64] but will not be further pursued in our work. Furthermore, non-universal couplings to h μν will be required in our case, in order to explain the already observed coupling in the V V (V = W, Z , g, γ ) channels, further spoiling the conservation of Tμν and thus gauge invariance [52, 65–67]. If one accepts the Pauli-Fierz Lagrangian coupled to the SM via (2.12) as a valid EFT in the linear regime, our paper then works to extrapolate the energy scale at which perturbative unitarity breaks down6 . This is to say that we are ignorant of the pathologies which appear beyond this bound, by setting it as a UV cutoff [50].

2.3 hZ → hZ In Sect. 1.2, it was pointed out that the SM Higgs boson uniquely moderates all the high energy scattering amplitudes. In the scenario where a spin-2 impostor mimics the current LHC signal, the coupling κi of the impostor to the W and Z bosons may be determined by the condition:  J =2 (h → V V ∗ ) =  J =0 (h S M → V V ∗ ).

(2.20)

We simply quote the result from our letter [23]: 2 = 3.16 × 10−5 GeV−2 , κW

κ Z2 = 4.42 × 10−5 GeV−2 .

(2.21)

The Feynman diagrams for the tree level elastic scattering h L L Z L → h L L Z L are shown in Fig. 2.1. Since the final states are distinguishable, only the s- and t-channels

5 We see in (2.19) that non-conserved currents can still be compatible with gauge invariance, so long

as this vanishes in the m h → 0 limit, as noted in [43]. level unitarity constraints have been studied in massive spin-2 theories [68].

6 Tree

2.3 h Z → h Z

31 

p2 , sρσ

k1 ,  α

β

k1 , α



β

p2 , sρσ

γ

p1 , sμν

k2 ,δ

p1 , sμν

γ

k2 ,δ

Fig. 2.1 Feynman diagrams depicting the scattering of the spin-2 impostor (double wavy lines) against the weak bosons (single wavy lines)

contribute. We now inspect the momentum dependences of various components of the Feynman diagram. Firstly, the appropriate h Z Z Feynman rule may be obtained in [69]:

(2.22) where: Cμν,αβ := ημα ηνβ + ημβ ηνα − ημν ηαβ ,  Dμν,αβ (k1 , k2 ) := ημν k1β k2α − ημα k1β k2ν + ημβ k1ν k2α − ηαβ k1μ k2ν + (μ ↔ ν) . (2.23) Then, by working in the rest frame of the propagator, the momenta of the incoming spin-1 and spin-2 state may be expressed as: μ

k1 ≈ p(1, 0, 0, 1), μ p1 ≈ p(1, 0, 0, −1),

(2.24)

respectively. The coordinate system can be chosen such that the final states lie in the x − z plane, and that the outgoing spin-1 and spin-2 momenta are respectively given by: μ k2 ≈ p (1, sin θ, 0, cos θ ) , (2.25) μ p2 ≈ p (1, − sin θ, 0, − cos θ ) , with θ measuring the scattering angle. Furthermore, the computation the scattering amplitude requires a specification of the polarisations vectors of the Z boson, which

32

2 Spin Determination of the LHC Higgs-Like Resonance

have to satisfy:  μ ( p, λ) pμ = 0, μ ( p, −λ) = μ∗ ( p, λ),

 μ ( p, λ)μ ( p, λ) = 0,  μ ( p, λ)μ ( p, −λ) = −1,

(2.26)

where λ gives the corresponding helicity. The polarisation vectors for the incoming spin-1 states are then:

μ0

≈

μ±

=

p mV p mV

(1, 0, 0, −1), incoming, (1, − sin θ, 0, − cos θ ), outgoing,

√1 (0, −1, ∓i, 0), 2 √1 (0, − cos θ, ∓i, sin θ ), 2

incoming,

(2.27)

outgoing.

On the other hand, the polarisation vectors for the spin-2 impostor read:  

1

1

1 s μν = μ+ ν+ , √ μ+ ν0 + μ0 ν+ , √ μ+ ν− + μ− ν+ − 2μ0 ν0 , √ μ− ν0 + μ0 ν− , μ− ν− , 2 6 2

(2.28) which are in turns constructed from the corresponding spin-1 polarisation vectors:

μ0

≈

μ± =

p (1, 0, 0, 1), mG p (1, sin θ, 0, cos θ ), mG

incoming, outgoing,

√1 (0, −1, ∓i, 0), 2 √1 (0, − cos θ, ∓i, sin θ ), 2

incoming,

(2.29)

outgoing.

One can summarise the momentum dependences of the various components of the Feynman diagram as follows: • • • •

Z longitudinal polarisation vector: O( p), h longitudinal polarisation tensor: O( p 2 ), h Z Z vertex: O( p 2 ), Z propagator: O(1).

√ From these, one naively expects that M(Z L h L L → Z L h L L ) ∼ O(s 5 ) where s is the centre of momentum energy. However, the calculation of the s- and t- channel processes yields a O(s 4 ) dependence (for more details, cf. Appendix A.2):

2.3 h Z → h Z

33

Fig. 2.2 The solid line shows the partial wave amplitude a0 and the dashed line the unitarised amplitude using the K -matrix formalism

32 p 8 κ 2 32 p 6 κ 2 m 2Z (1 − cos θ ) , Ms = − 4 + 2 3m G (s − m Z ) 3m 4G (s − m 2Z )    off diagonal

2 p 8 κ 2 (1 + cos θ )4 p 10 κ 2 csc2 θ2 sin6 θ , Mt = − − 3m 4G (t − m 2Z ) 3m 4G m 2Z (t − m 2Z )   

(2.30)

off diagonal

where terms from the off-diagonal part of the Z -propagator are indicated. The zeroth order partial wave was evaluated as per (1.48). It is clear from Fig. 2.2 that the unitarity bound (A.21) is violated at  ∼ 600 GeV. We proceed to see what additional resonances may restore perturbative unitarity (cf. Sect. 1.1.2). Whilst an addition resonance with J = 1, 2 or 3 are expected from naïve spin considerations, we restrict ourselves to the spin-1 case because the higher spin cases lead to amplitudes of higher momentum dependence and bring forward the onset of unitarity violation. Such a spin-1 Z  -boson is introduced by a kinetic-mixing term associated with the SM Z -boson: 1 μν L ⊃ − Z μν Z  , 2

(2.31)

The Feynman rule of the h Z Z  vertex can then be appropriately extracted as follows:  − iκ  (k Z · k Z  ) Cμν,ρσ + Dμν,ρσ (k Z , k Z  ) .

(2.32)

The mixing term then contributes to h L L Z L → h L L Z L scattering through the s- and t- channels amplitudes which read:

34

2 Spin Determination of the LHC Higgs-Like Resonance

Fig. 2.3 K -matrix formalism projects a real amplitude onto the Argand circle corresponding to (Appendix A.18)

Im

i 2

 −1 1 aK j (s) = Re aj (s) − i

aj (s)



8κ  2 p 6 m 2Z (1 − cos θ ) , 3m 4G (s − m 2Z  )   κ 2 p 6 csc θ2 sin4 θ p 4 sin2 θ + 2 p 2 m 2Z cos θ + m 4Z . =− m 4G m 2Z (t − m 2Z  )

Re

MsZ =  MtZ

(2.33)

The top two panels of Fig. 2.4 show the amplitudes with κ = κ  and the bottom, with κ = iκ  . The latter relation between the couplings is chosen in attempt to cancel the leading order momentum dependence. It can be observed that there is no symmetry present in the Z − Z  sector that results in the cancellation of the power law dependence. Subsequently, we investigated whether this problem can be cured with another unitarisation method. The K -matrix ansatz [70–72] formally corresponds to adding infinitely heavy and wide resonances which unitarise the amplitude by projecting it onto the Argand circle as shown in Fig. 2.3. In relation to this, [73] showed that the amplitude from the higher order expansion of a chiral perturbation theory converges towards the unitary circle and that the inverse amplitude methods give a good parametric description. It was subsequently argued that when tree-unitarity violation is lower than the onset of new physics, a chiral theory will heal itself with a resonant structure with scalar quantum numbers. In the context of the electroweak sector, this corresponds to the introduction of the Higgs. The saturation observed in Fig. 2.4 beyond the Z  resonance is due to unitarisation and indicates that a UV completion of the theory is required to correctly extrapolate the physics in this region [74]. These results demonstrate that a theory describing the 125 GeV resonance as a spin-2 Higgs impostor is ill from a unitarity viewpoint. Furthermore, violation of perturbative unitarity cannot be amended by adding further resonances within the weakly coupled regimes of the theory.

2.3 h Z → h Z

35

0.4

Z’

m no

mZ ’ 584 GeV

a0

0.6

mZ ’ 500 GeV

no

mZ

’

0.8

mZ ’ 400 GeV

mZ ’ 584 GeV

Re a0

10

mZ ’ 500 GeV

mZ ’ 400 GeV

1.0 15

5 0.2 0 300

400

500

600

700

0.0 300

800

400

s GeV

500

600

700

800

s GeV

Z’

mZ ’ 600 GeV no m

a0

0.6

mZ ’ 500 GeV

Z’

0.8

no m

mZ ’ 600 GeV

6

mZ ’ 500 GeV

Re a0

8

1.0 mZ ’ 400 GeV

10

mZ ’ 400 GeV

12

4

0.4

2

0.2

0 300 400 500 600 700 800 900 1000

0.0 300 400 500 600 700 800 900 1000

s GeV

s GeV

Fig. 2.4 The tree level h Z → h Z scattering amplitude with various masses of Z  added to the theory, assuming κ = κ  (top) and κ = iκ  (bottom). The corresponding unitarised amplitudes are shown on the right. Source [23]

2.4 ST U Parameters The contribution of the impostor to the ST U parameters (cf. Sect. 1.2.2) is due to the rainbow diagram as shown in Fig. 2.5. We simply adapt the calculation done in [75] and remove the contributions from the massive K K tower [75]. This is detailed in our Letter and will not be reproduce it here. Instead, we show some representative values of such modifications in Table. 2.1 and revised the associated figure from our later in Fig. 2.6. This makes clear that the measured oblique parameters cannot be made consistent with the measured value (cf. Ref. [76]) via removal of the SM Higgs contributions [77]:

k

Fig. 2.5 Rainbow diagram

p

p+k

p

36

2 Spin Determination of the LHC Higgs-Like Resonance

Fig. 2.6 Contribution of the spin-2 impostor and the SM Higgs to the ST U oblique parameters. It is clear here that the impostor is not compatible with the experimental values

Table 2.1 Representative values of S, T and U parameters in the spin-2 Higgs impostor effective theory. Source [23] Parameter/Energy 200 GeV 400 GeV 600 GeV cut-off () S T U

−3.11 −0.698 −0.163

Sh S M Th S M

−58.0 −9.75 −3.38

−426.0 −64.6 −26.5

   1 ln = , ref 6π mh

  3  =− ln , ref 8π cos2 θW mh

(2.34)

Uh S M = 0, and replacement with those of a spin-2 impostor. It should be noted that here  denote the new physics scale expected at the unitarity violation scale ∼600 GeV (cf. Sect. 2.3). The corrections to ST U parameter due to the spin-2 impostor are sufficiently larger than those in [75] since the K K masses are sufficiently larger than that of the 125 GeV impostor (m iK K  m h ), and that these masses appear in the denominator of the self energy calculations.

2.5 Remarks The argument against a minimally coupled spin-2 based on unitarity was generalised for more generic spin-2 couplings in our work [23]. Table.1 of [17] lists the most general tensor couplings for h Z Z ∗ . The momentum dependences in the corresponding

2.5 Remarks

37

amplitudes are the same, if not higher than the minimal case presented in this chapter, unless the coefficients are carefully tuned. We concluded that unitarity violation is expected below   1 TeV. This is applicable whenever the SM scalar is replaced with a spin-2 boson assuming generic couplings tuned to mimic the h S M → W W and h S M → Z Z decay rates. Furthermore, the resulting couplings between the spin2 boson and the electroweak bosons cause severe disagreement with the measured ST U precision parameters. The existence of a spin-2 Higgs impostor then cannot be accommodated in an effective theory and be made compatible with experiments.

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20. S. Bolognesi, Y. Gao, A.V. Gritsan, K. Melnikov, M. Schulze, N.V. Tran et al., On the spin and parity of a single-produced resonance at the LHC. Phys. Rev. D 86, 095031 (2012). arXiv:1208.4018 21. F. Maltoni, K. Mawatari, M. Zaro, Higgs characterisation via vector-boson fusion and associated production: NLO and parton-shower effects. Eur. Phys. J. C 74, 2710 (2014). arXiv:1311.1829 22. F. Demartin, F. Maltoni, K. Mawatari, B. Page, M. Zaro, Higgs characterisation at NLO in QCD: CP properties of the top-quark Yukawa interaction. Eur. Phys. J. C 74, 3065 (2014). arXiv:1407.5089 23. A. Kobakhidze, J. Yue, Excluding a generic spin-2 Higgs impostor. Phys. Lett. B 727, 456–460 (2013). arXiv:1310.0151 24. ATLAS collaboration, Study of the spin of the new boson with up to 25 fb−1 of ATLAS data, ATLAS-CONF-2013-040 25. CMS collaboration, Combination of standard model Higgs boson searches and measurements of the properties of the new boson with a mass near 125 GeV, CMS-PAS-HIG-13-005 26. L.D. Landau, On the angular momentum of a system of two photons. Dokl. Akad. Nauk Ser. Fiz. 60, 207–209 (1948) 27. C.-N. Yang, Selection rules for the dematerialization of a particle into two photons. Phys. Rev. 77, 242–245 (1950) 28. M. Cacciari, L. Del Debbio, J. R. Espinosa, A. D. Polosa, M. Testa, A note on the fate of the Landau-Yang theorem in non-Abelian gauge theories. arXiv:1509.07853 29. W. Beenakker, R. Kleiss, G. Lustermans, No Landau-Yang in QCD. arXiv:1508.07115 30. AYu. Ignatiev, G.C. Joshi, M. Matsuda, The search for the decay of Z boson into two gammas as a test of Bose statistics. Mod. Phys. Lett. A 11, 871–876 (1996). arXiv:hep-ph/9406398 31. S.N. Gninenko, AYu. Ignatiev, V.A. Matveev, Two photon decay of Z’ as a probe of Bose symmetry violation at the CERN LHC. Int. J. Mod. Phys. A 26, 4367–4385 (2011). arXiv:1102.5702 32. V. Pleitez, The angular momentum of two massless fields revisited. arXiv:1508.01394 33. J. P. Ralston, The need to fairly confront spin-1 for the new Higgs-like Particle. arXiv:1211.2288 34. S. Moretti, Variations on a Higgs theme. Phys. Rev. D 91, 014012 (2015). arXiv:1407.3511 35. A. Koenigstein, F. Giacosa, D. H. Rischke, Classical and quantum theory of the massive spin-two field. arXiv:1508.00110 36. K. Hinterbichler, Theoretical aspects of massive gravity. Rev. Mod. Phys. 84, 671–710 (2012). arXiv:1105.3735 37. C. de Rham, Living Rev. Rel. 17, 7 (2014). arXiv:1401.4173 38. S.F. Hassan, R.A. Rosen, Resolving the ghost problem in non-linear massive gravity. Phys. Rev. Lett. 108, 041101 (2012). arXiv:1106.3344 39. K. Hinterbichler, R.A. Rosen, Interacting spin-2 fields. JHEP 07, 047 (2012). arXiv:1203.5783 40. S. Folkerts, A. Pritzel, N. Wintergerst, On ghosts in theories of self-interacting massive spin-2 particles. arXiv:1107.3157 41. L.P.S. Singh, C.R. Hagen, Lagrangian formulation for arbitrary spin. 1. The boson case. Phys. Rev. D 9, 898–909 (1974) 42. L.P.S. Singh, C.R. Hagen, Lagrangian formulation for arbitrary spin. 2. The fermion case. Phys. Rev. D 9, 910–920 (1974) 43. G. Dvali, O. Pujolas, M. Redi, Non Pauli-Fierz massive gravitons. Phys. Rev. Lett. 101, 171303 (2008). arXiv:0806.3762 44. M. Fierz, W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field. Proc. Roy. Soc. Lond. A 173, 211–232 (1939) 45. W. Pauli, M. Fierz, On Relativistic Field Equations of Particles With Arbitrary Spin in an Electromagnetic Field. Helv. Phys. Acta 12, 297–300 (1939) 46. P. Van Nieuwenhuizen, On ghost-free tensor lagrangians and linearized gravitation. Nucl. Phys. B 60, 478–492 (1973)

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47. E.C.G. Stueckelberg, Interaction forces in electrodynamics and in the field theory of nuclear forces. Helv. Phys. Acta 11, 299–328 (1938) 48. R. Rahman, Higher Spin Theory - Part I. PoS ModaveVIII, 004 (2012). arXiv:1307.3199 49. A. Peach, Dissertation higher-spin gauge theories, Vasiliev theory and holography, Master’s thesis, Durham University, 2013 50. M. Porrati, R. Rahman, Intrinsic cutoff and acausality for massive spin 2 fields coupled to electromagnetism. Nucl. Phys. B 801, 174–186 (2008). arXiv:0801.2581 51. D.G. Boulware, S. Deser, Classical General Relativity Derived from Quantum Gravity. Annals Phys. 89, 193 (1975) 52. M.D. Schwartz, Quantum Field Theory and the Standard Model, (Cambridge University Press, 2014) 53. N. Arkani-Hamed, H. Georgi, M.D. Schwartz, Effective field theory for massive gravitons and gravity in theory space. Annals Phys. 305, 96–118 (2003). arXiv:hep-th/0210184 54. S. Deser, Selfinteraction and gauge invariance. Gen. Rel. Grav. 1, 9–18 (1970). arXiv:gr-qc/0411023 55. R.M. Wald, Spin-2 fields and general covariance. Phys. Rev. D 33, 3613 (1986) 56. T. Ortin, Gravity and Strings. Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2015) 57. H. van Dam, M.J.G. Veltman, Massive and massless Yang-Mills and gravitational fields. Nucl. Phys. B 22, 397–411 (1970) 58. V.I. Zakharov, Linearized gravitation theory and the graviton mass. JETP Lett. 12, 312 (1970) 59. P. Creminelli, A. Nicolis, M. Papucci, E. Trincherini, Ghosts in massive gravity. JHEP 09, 003 (2005). arXiv:hep-th/0505147 60. A.I. Vainshtein, To the problem of nonvanishing gravitation mass. Phys. Lett. B 39, 393–394 (1972) 61. D.G. Boulware, S. Deser, Can gravitation have a finite range? Phys. Rev. D 6, 3368–3382 (1972) 62. C. de Rham, G. Gabadadze, Generalization of the Fierz-Pauli action. Phys. Rev. D 82, 044020 (2010). arXiv:1007.0443 63. C. de Rham, G. Gabadadze, A.J. Tolley, Resummation of massive gravity. Phys. Rev. Lett. 106, 231101 (2011). arXiv:1011.1232 64. C. de Rham, G. Gabadadze, Selftuned massive spin-2. Phys. Lett. B 693, 334–338 (2010). arXiv:1006.4367 65. S. Weinberg, E. Witten, Limits on massless particles. Phys. Lett. B 96, 59 (1980) 66. S. Weinberg, Infrared photons and gravitons. Phys. Rev. 140, B516–B524 (1965) 67. A. Jenkins, Constraints on emergent gravity. Int. J. Mod. Phys. D 18, 2249–2255 (2009). arXiv:0904.0453 68. N. D. Christensen, Stefanus, On tree-level unitarity in theories of massive spin-2 bosons. arXiv:1407.0438 69. T. Han, J.D. Lykken, R.-J. Zhang, On Kaluza-Klein states from large extra dimensions. Phys. Rev. D 59, 105006 (1999). arXiv:hep-ph/9811350 70. A.M. Badalian, L.P. Kok, M.I. Polikarpov, YuA Simonov, Resonances in coupled channels in nuclear and particle physics. Phys. Rept. 82, 31 (1982) 71. W. Kilian, T. Ohl, J. Reuter, M. Sekulla, High-energy vector boson scattering after the Higgs discovery. Phys. Rev. D 91, 096007 (2015). arXiv:1408.6207 72. S.U. Chung, J. Brose, R. Hackmann, E. Klempt, S. Spanier, C. Strassburger, Partial wave analysis in K matrix formalism. Ann. Phys. 4, 404–430 (1995) 73. U. Aydemir, M.M. Anber, J.F. Donoghue, Self-healing of unitarity in effective field theories and the onset of new physics. Phys. Rev. D 86, 014025 (2012). arXiv:1203.5153 74. A. Alboteanu, W. Kilian, J. Reuter, Resonances and unitarity in weak boson scattering at the LHC. JHEP 11, 010 (2008). arXiv:0806.4145 75. T. Han, D. Marfatia, R.-J. Zhang, Oblique parameter constraints on large extra dimensions. Phys. Rev. D 62, 125018 (2000). arXiv:hep-ph/0001320

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Chapter 3

Probing CP -violating Top-Yukawa Couplings at the LHC

Having established confidently that the spin of h(125) should be J = 0, the next focus is to pin down its CP-properties. If this scalar resonance is a CP-eigenstate, it may either be even (scalar) or odd (pseudoscalar). However, it may well be that it is not a CP-eigenstate, indicating the existence of CP-violation in the Higgs sector (cf. e.g. [1]). One must therefore study the couplings of h to the gauge bosons and fermions in order to elucidate its role in electroweak symmetry breaking. The final step is to reconstruct the form of the Higgs potential, through measurements of the self-couplings. This demands the other SM couplings to be measured at high precision. The aim of this chapter is to motivate and explain the works [2, 3]. We begin with Sect. 3.1, which shows that a non-linear realised electroweak symmetry may give a CP-violating Yukawa sector: Lpheno ⊃ − f (y Sf + i y Pf γ 5 ) f h.

(3.1)

Such couplings induce Higgs couplings to the bosons V = W , Z , γ and g through fermion loops, which read: L⊃

 α  cV V μν Vμν + c˜V V μν V˜μν h, 8π v

(3.2)

where V˜ μν := 21  μνρσ Vρσ . The Low Energy Theorem will also be invoked to understand the origins of these couplings. Naïvely, the W and Z tree level terms  2 κV mvV hV μ Vμ are more likely to favour the SM hypothesis—the CP-odd contributions can only arise at loop-level and are suppressed by a relative factor of 2  c˜V αw v . Particularly, the exclusion limits of the J P = 0− hypothesis at 97.8% κV m V C.L. by ATLAS [4], 97.6% C.L. (and later >99.9% C.L.) by CMS [5, 6] are established in the h → Z Z ∗ → 4 channel alone. This is done in a similar way to the J = 2 exclusion based on kinematic variables [7–12]. The fermion sector is likely to offer better prospects to investigate the CP-structure of the Higgs resonance. We show in Sect. 3.1, that such CP-odd contributions can © Springer International Publishing AG 2017 J.T.S. Yue, Higgs Properties at the LHC, Springer Theses, DOI 10.1007/978-3-319-63402-9_3

41

42

3 Probing CP -violating Top-Yukawa Couplings at the LHC

be motivated from dimension-six operators or from our non-linearly realised gauge symmetry, giving y Pf /y Sf ∼ O(1). At the LHC, gg → h → bb, τ + τ − events are produced in abundance, but the extraction of physics from these channels is made difficult by substantial QCD backgrounds. However, τ τ decay channel have still received much attention since the decay products preserve spin information of the τ -lepton [13]. New physics effects are expected to be more pronounced with the top quark given that its coupling with the Higgs is the strongest in the SM. Unitarity considerations suggests that the scale of top mass generation has to be  3 TeV [14–18]. Furthermore, Yukawa couplings tend to destabilise the electroweak vacuum (cf. e.g. [19–21]). Such couplings also play an important role in CP-violating scatterings, which enables electroweak baryogenesis (cf. Chap. 4). We were therefore motivated to study the top-Higgs sector in [2, 3]. Particularly, a CP-violating Yukawa sector will modify the kinematics, along with the production and decay ratesof the Higgs boson. In Sect. 3.2, we first check the compatibility with the measured production and decay rate from LHC, as well as the electron dipole moments. However, the modified gg → h and h → γ γ rates1 only provide indirect constraints to such CP-violating top-Yukawa couplings. It is therefore important to carry out collider studies of the pp → tth and pp → th j channels. As will be explained in subsequent parts of this chapter, these production modes are the leading processes which allow us to directly probe the tth couplings. Unlike all other fermions, a direct implication of the heavy top mass is that the Higgs cannot decay to a top quark pair on-shell, since m t > m h /2. As a result, studies have focused on Higgs production associated with a top pair, pp → tth, where the top-Higgs coupling first enters at tree level (see e.g. [25–30]). The prospects of distinguishing the scalar and pseudoscalar component of the tth couplings at the LHC have been proposed in [31–38]. After the 8 TeV run, an upper limit on the signal strength, κt ∈ [−1.3, 8.0] has been given by the ATLAS Collaboration at 95% C.L. limit, using the h → γ γ modes [39]. A corresponding limit of μ < 3.3 is found in the bb channel [40]. The t-quark is also unique as it is the only quark that decays before it hadronises, with spin information inherited by its decay products [41–54]. We consider in Sect. 3.3, angular correlations in the t-decay product as a consequence of top quark polarisation. Such phenomena have been exploited frequently (see e.g. [33, 36, 55– 60]) to probe for CP-violation in top-Yukawa couplings. Likewise, anomalous chiral structures can be probed in single top processes as they readily produce polarised top quarks [61–70]. We were led to consider pp → th j production in [2, 3], and in Sect. 3.4, the relevant collider phenomenology will be addressed. This channel is subsequently shown to be promising, as CP-violating tth couplings enhance the production cross section. The amount of CP-violation may then be probed at the LHC via the asymmetries in the spin-correlations of the final state leptons. Section 3.4 will discuss [2], where the h → bb decay mode was found to be difficult due to large The probing of CP -violating tth couplings has been considered in h → γ γ decays alone in [22–24].

1

3 Probing CP -violating Top-Yukawa Couplings at the LHC

43

QCD backgrounds, despite having a large branching ratio. Subsequently, the work [3] is also presented, where h → γ γ is shown to be outperform the previous mode, because of the additional enhancement in the decay rate, and clean invariant mass reconstruction. We conclude this chapter with some remarks in Sect. 3.5.

3.1 Non-linear Realisation in the Top-Higgs Sector The LHC has not completely established the electroweak symmetry breaking mechanism. A CP-violating Yukawa sector is allowed in the non-linear realisation of the electroweak symmetry (cf. Sect. 1.2). We first reparameterise the associated renormalisable Higgs-fermion interactions from (1.56) as follows:  LY ⊃ −

 m i(u) j

   yi(u) yi(d) i i j j j (d)  ˜ Rj + √ ρ Q L u R − m i j + √ ρ Q L d 2 2   yi( ) i j j ( )  − m i j + √ ρ L L R + h.c., 2

(3.3)

i a3 a a ˜ := iσ 2 ∗ , with σ a denoting the Pauli where := e− 2v (Iδ −σ )π (0, 1)T and matrices for a = 1, 2, 3. We note that auxiliary mass parameters m i j are allowed. As explained previously, much focus is placed on the top quark because new physics effects in the symmetry breaking sector are more likely to manifest through its large coupling to the Higgs. We therefore take all parameters in the Lagrangian (3.3) to be that of the SM values, with the exception of those corresponding to the t-quark (which now also has an auxiliary mass m t ):



m i(d, ) = 0, j yi(d, ) = yS(d, ) j M ij,



 m i(u) j = diag(0, 0, m t ), √  √ 2m u 2m c (u) , , yt eiξ . yi j = diag v v

(3.4)

When the field ρ acquires its vacuum expectation value ρ = v, the physical Higgs field may be identified through: ρ(x) = v + h(x). The Lagrangian then becomes:

(3.5)

3 Probing CP -violating Top-Yukawa Couplings at the LHC

44

yt  L ⊃ m t eiξ t L t R + √ heiξ t L t R + h.c., 2 

2

2 yt yt 2  m t + √ v cos ξ + √ v sin ξ , m t := 2 2 sin ξ vy t tan ξ  := √  . 2m t + yt v cos ξ

(3.6)



After field rotation t R → e−iξ t R to obtain the physical top field with real mass, the interaction term is then described by (3.1) with: √ yt yt ( 2m t cos ξ + yt v)  = √ cos(ξ − ξ ) = , 2m t 2 yt yt m t sin ξ ytP = √ sin(ξ − ξ  ) = . 2m t 2 ytS

(3.7)

In line with [2, 3], it will be assume in the remainder of this chapter (except Sect. 3.2.2), that such rephasing is already absorbed into the √ definition of ξ . The SM case may be recovered when yt takes the value ytS M := 2m t /v, and ξ = 0. The vacuum expectation of the Higgs field will be continued to be denoted by v ≈ 246 GeV. A Lagrangian assuming the form in (3.1) can also be motivated from dimensionsix effective operators [71–75] (for examples of renormalisable models, see e.g. [76]):

H†H Ldim≤6 ⊃ − α + β 2 H Q †L t R + h.c., 

(3.8)

where α, β ∈ C are dimensionless parameters and , the new physics scale. After √ symmetry breaking with H = (0, v + h/ 2)T , it may be identified that: v2 v2 yt eiξ = α + β 2 +2β 2 .    

(3.9)

ytS M

Assuming that new physics enters at the TeV scale ( ∼ 10v) and |β| ∼ O(1), it is conceivable that the phase ξ may assume the full range (−π, π ] [13].

3.1.1 Contribution to Loops We now shift our attention to the loop induced hV V couplings which readily follows by integrating out the propagating particles in the loop (cf. Feynman diagrams in

3.1 Non-linear Realisation in the Top-Higgs Sector

45

γ(k1 )

γ(k1 )

−→

h(p)

h(p)

γ(k2 )

γ(k2 )

Fig. 3.1 Effective vertex by integrating out particles in a loop

Fig. 3.1). Here we show only the case for V = γ with fermions loops, as a similar procedure follows for the V = g case and is presented in [77]. Beginning with the Feynman rule corresponding to (3.2): −i

  αem   μ ν cγ k1 k2 − k1 · k2 ημν + c˜γ  μνρσ k1ρ k2σ , 2π v

(3.10)

the coefficients cγ and c˜γ can be extracted by matching the known decay rates (cf. Eq. 3.22) and then sending m t → ∞. The results will be shown to be consistent with the Low Energy Theorem (LET) [78–83], which states that: 1 lim M(h X ) = ph →0 v



∂ ∂ ln m f

M(X ).

(3.11)

This relates the amplitude of a process X such as γ → γ , to one that involves an extra Higgs external leg, whose amplitude is then M(h X ) ∼ M(h → γ γ ). For the example just given, it is a statement that the hγ γ interaction reproduces the photon two-point function in the soft limit. It is trivial to see this if the Higgs is attached to a bosonic propagator. However, if the Higgs is attached to a fermionic propagator, the result then follows from the new propagator given by: i pi − m i 

→

i −im i i i mi ∂ = . pi − m i v  pi − m i pi − m i v ∂m i  

(3.12)

To use the LET, first start with the γ → γ process described by the renormalised Lagrangian: ⎛ ⎞ ⎜ ⎟  α ⎜ 1 2 ⎟ ⎟, 1 + L ⊃ − F μν Fμν ⎜ b ln s ⎜ 4 4π i m i2 ⎟ ⎝ i  ⎠ =iγ γ (0)

(3.13)

3 Probing CP -violating Top-Yukawa Couplings at the LHC

46

where iγ γ (0) are the loop contributions to the zero-energy vacuum polarisation function (cf. Eq. 1.52) due to the species i (with spin si ), with b1 = −7 and b1/2 = 4 N Q 2 . Straightforward application of the LET then gives the scalar part: 3 c i

∂ α μν F Fμν bsi ⊃ ln m i . 8π v ∂ ln m i

Lscalar hγ γ

(3.14)

The generalisation to pseudoscalar couplings can be found in [81, 84], which relies on the Adler-Bell-Jackiw (ABJ) anomaly2 [85, 86]:   Nc Q 2f α μν F F˜μν . ∂μ f γ μ γ5 f = 2im f f γ5 f + 2π  

(3.15)

μ

= j5

This relation is not modified by radiative corrections due to the Adler-Bardeen theorem [87]. The pseudoscalar part of (3.2) then follows directly from [88]: μ

lim h|∂μ j5 h|γ γ = 0, =⇒

ph →0

pseudoscalar

Lhγ γ

2Nc Q 2f α

⊃

8π v

F μν F˜μν .

(3.16)

We note that the form factors in (A.25) take the limits:  cγ =

2Nc Q 2f 

c˜γ = 2Nc Q 2f

y Sf y Sf M y Pf y Sf M

 Fs (τ )  F p (τ )





7 2 τ + O(τ ) , 30 y Sf M  S 

yf 1 2 2 1 + τ + O(τ ) , = 2Nc Q f 3 y Sf M (3.17) 4 = Nc Q 2f 3

y Sf

1+

which agree with the t-quark contributions to the scalar (3.14) and pseudoscalar (3.16) effective Lagrangian. Now that the origins of CP-violating couplings of h to the bosons and fermions are clear, constraints on these couplings can be discussed.

3.2 Bounds on CP-violating Couplings 3.2.1 Branching Ratios and Production Cross Sections It is generically not practical to fit all the Higgs couplings as free parameters. Instead, the κ-framework [89, 90] is widely used in global fits, where the couplings are scaled from the respective SM values via the coupling strength modifiers κi . The relevant modifiers for the top-Yukawa couplings are defined as follows: 2 This

is also related to baryon number violation which will be explained in Sect. 4.4.

3.2 Bounds on CP -violating Couplings

κtS :=

yt cos ξ, ytS M

47

κtP :=

yt sin ξ, ytS M

(3.18)

whilst the other tree-level couplings are kept as per SM. The observables are then the signal strengths μi given by the signal rate of the mode i normalised to that of the SM:  σ (Pi )B(Di ) , (3.19) μi :=  i i σ S M (Pi )B S M (Di ) where σ (Pi ) is the cross section of the production mode of i and B(Di ) is the branching ratio for the decay mode of i. The dominant process through which the Higgs is produced is via gluon fusion gg → h. A Higgs mass of 125 GeV allows for a plethora of decay modes. Although the leading decay channels are h → bb followed by h → W W ∗ , the 2012 Higgs discovery was made in the h → γ γ and h → Z Z ∗ → 4 decay channels, where clean signatures led to a good mass reconstruction. Both the gg → h and h → γ γ processes are loop mediated by the t-quark and so can be used to indirectly constrain the top-Yukawa couplings. In particular, the partonic cross-section of gluon fusion is related to the inverse Higgs decay to two gluons in the narrow width approximation: σˆ (gg → h) =

π2 (h → gg). m 3h

(3.20)

ATLAS and CMS have already obtained the following constraints (cf. [91]): κtS ∈ [−1.2, −0.6] ∪ [0.6, 1.3],

ATLAS,

κtS

CMS.

∈ [0.6, 1.2],

(3.21)

In the SM, the decay width into γ γ , Z γ and gg are proportional to the squared scalar form factor. Under the assumption that no beyond the SM particles propagate in the loops, the rates have additional squared pseudoscalar form factors [92, 93]: m 3h αw2 (|S γ (m h )|2 + |P γ (m h )|2 ), 256π 3 v 2 m 3h αs2 (|S g (m h )|2 + |P g (m h )|2 ), (h → gg) = 32π 3 v 2

2 αw m 2W sW m 2Z 3 Z γ 3 (h → Z γ ) = 1 − m (|S (m h )|2 + |P Z γ (m h )|2 ). 128v 4 π 4 h m 2h (h → γ γ ) =

(3.22)

The analytical expressions for the form factors are taken from [94] and are collected in Appendix A.3. Neglecting the contributions from quarks other than that of the top quark, the form factor may be approximated as:

3 Probing CP -violating Top-Yukawa Couplings at the LHC

48

S

S γ (m h ) ≈ −8.32κW + 1.83κtS ,

P γ (m h ) ≈ 2.79κtP ,

S g (m h ) ≈ 0.688κtS ,

P g (m h ) ≈ 1.048κtP ,

Zγ

(m h ) ≈ −11.8695κW +

0.6426κtS

P

Zγ

(m h ) ≈

(3.23)

0.9751κtP .

Three important remarks are in order. Firstly, the lack of interference terms between the scalar and pseudoscalar parts in (3.22) are due to the orthogonality between the identity and γ 5 structure in the respective couplings. Secondly, it is obvious from (3.23) that the t-loop factors are larger for the pseudoscalar than that of the scalar component. Lastly, given the above considerations and the fact that W -contribution interferes destructively with the t-contribution, the rates will be enhanced with increasing ξ . We present results updated from [2], where we profile the likelihood based on a chi-square fit: ˆ T C(µ − µ), ˆ (3.24) χ 2 = (µ − µ) using the package Lilith-1.1.3 [95]. Here, C is the covariance matrix taking into account the correlations between the theoretical cross section, branching ratio and luminosity uncertainties amongst the different observables. The database contains the ATLAS and CMS Run I data up to September of 2015. The bounds on the scalings of the scalar and pseudoscalar top Yukawa couplings, κtS and κtP , can then be read off from Fig. 3.2. At 95% C.L., the new constraints are: κtS ∈ [0.65, 1.25],

κtP ∈ [−0.55, 0.55].

(3.25)

Of course, each of the above constraints assumes that the remaining parameters are already marginalised. Correspondingly, the allowed range of the magnitude yt /ytS M ∈ [0.90, 1.23] at ξ = 0 shrinks and decreases to yt /ytS M ∈ [0.8, 0.9] at |ξ | = 0.2π . Deviations from the SM in the hgg, hγ γ and h Z γ rates are parameterised in terms of:

Fig. 3.2 Constraints on κtS and κtP (left), or equivalently, yt /ytS M and ξ (right). The solid and dashed contours represent the 68% and 95% C.L. limits respectively

3.2 Bounds on CP -violating Couplings

49

Fig. 3.3 Fit of the effective hγ γ , hgg and h Z γ couplings (cf. Eq. 3.26) over ξ , which are marginalised over yt . The projected sensitivity of HL-LHC is shown by the hatched region for Cγ γ and C gg only, as the current constraints have already surpassed the 10% precision on C Z γ

C hgg

        γ 2  g2   S Z γ 2 +  P Z γ 2  |S | + |P γ |2  |S | + |P g |2  :=   g 2 , C hγ γ :=  , C := ,    γ 2 hZγ   Z γ 2 S  S  SM SM SS M 

(3.26) The constraints on these reduced couplings are shown in Fig. 3.3 with ξ , along with the expected precision reached at the HL-LHC (High Luminosity LHC). One observes a 95% C.L. limit of ξ ∈ [−0.2π, 0.2π ], a marked improvement from the previous bound ξ ∈ [−0.6π, 0.6π ] from [2].

3 Probing CP -violating Top-Yukawa Couplings at the LHC

50

3.2.2 EDM Constraints The new source of CP-violation in the Higgs-top sector induces additional contributions to the electric dipole moments (EDMs) of charged fermions, d f , due to the two-loop Barr-Zee type diagram [96] as Fig. 3.4). For the electron EDM, the contribution is given by (see e.g. [97, 98]):  2

2  mt mt 16 α de me S P P S y + y , = y f y f e t 1 e t 2 2 S M 3 S M 2 e 3 (4π ) yt ye v mh m 2h

(3.27)

where the loop functions f 1,2 are given by: f 1 (x) := √



2x 1 − 4x



Li2 1 −

1−

   √ √ 1 − 4x 1 + 1 − 4x − Li2 1 − , 2x 2x

f 2 (x) := (1 − 2x) f 1 (x) + 2x(ln x + 2), ! x dz ln(1 − z). Li2 := − z 0 (3.28)   Using the limit f 1 (z) → ln z + 2 + O √1z , one estimates the contribution from our top-Higgs sector to be: de /d e =

2 sin ξ 3



m t mt



yt



ytS M

ln

m 2t m 2h



≈ 0.22 sin ξ

yt ytSM



m t mt

,

(3.29)

|e|αm e −27 where d e = 16π e·cm, and ytS M is the SM top-Yukawa coupling. 3 v 2 ≈ 2.5 · 10 We stress that the considerations here differs from those in e.g. [38, 97, 99–105] due to the extra factor of m t /m t . The origin of this factor is that m t is an additional parameter in a non-linearly realised Higgs sector (cf. Sect. 3.1). It is clear that de → 0

γ, Z, g

γ, Z, g

t

W± γ, g

h f

γ, g

h f

Fig. 3.4 Two-loop Barr-Zee diagrams leading to contributing to the electron EDM

3.2 Bounds on CP -violating Couplings

51

as m t → 0 because in this limit, the CP-phase in the mass term is then aligned with that of the Yukawa coupling. In such a scenario, after a field redefinition of t R (cf. Eq. 3.6) to go into the physical basis, there is no CP-violation (at least in the diagonal basis that we have assumed). Due to the extra suppression of sin ξ and m t /m t , most of the parameter space allowed by the Higgs data obtained from the fit in the last section (Sect. 3.2.1) overlaps with the eEDM constraints obtained by the ACME collaboration using thorium monoxide (ThO) [106]: deex p < 8.7 × 10−29 e · cm, (ThO).

(3.30)

Finally, we note that these bounds may be easily evaded since the light quark Yukawa couplings are not directly observable and accidental cancellations in the eEDM are possible with anomalies in other Yukawa couplings. These constraints will not be further imposed in our model.

3.3 Polarisation Phenomenology The top quark is the most massive quark and is often considered as a sensitive probe for electroweak physics. Unlike the lighter quarks, the electroweak decay of the tquark occurs before hadronisation and subsequently its polarisation can be inferred from its decay products without significant contamination from QCD. In Sect. 3.3.1, we discuss the origins of the correlation between the top spin direction and its decay product’s emission direction as due to the V − A structure of the weak coupling. The CP-violating top Yukawa coupling introduces right-handed couplings and will change this differential distribution. Given that the top’s decay width is sufficiently smaller than the top mass t  m t , the amplitude for a process such as pp → t X → W bX is effectively factored over production and decay processes through the narrow width approximation (cf. Sect. 3.3.2). A full calculation of the cross section is made difficult by interference terms between the spin up and spin down tops (see e.g. [68]). However, if the produced tops are polarised, this can be simplified via a suitable choice of quantisation axis [48– 54, 107, 108] (see [109] specifically for single top processes). Particularly, ATLAS and CMS [110, 111] have studied theses effects in single top production (without a Higgs). This channel is important as it is the leading channel which produces polarised t-quarks at the LHC. The polarisation phenomenology in this channel will be first considered since there are many resemblances to be found in the pp → th j process of Sect. 3.4.

3 Probing CP -violating Top-Yukawa Couplings at the LHC

52

3.3.1 Lepton Spin-Correlation The aim of this section is then to show that the lepton from t → + ν b is strongly correlated to the spin-axis of the top. Subsequently, the result is generalised also for the production of the t-quark in Sect. 3.3.2. The amplitude for Fig. 3.5 is given by:      −i gμν − kμ kν /m 2W −ig2 μ −ig2 μ u ν √ γ P− v , (3.31) M = i u b Vtb √ γ P− u t k 2 − m 2W + i MW W 2 2 where k = t − b. To facilitate the calculation we follow the discussion in [112] and employ the spin-helicity formalism. One has to first introduce two massless momenta: 1 t± = (t ± m t st ) , (3.32) 2 where t is the four-momentum of the t-quark and st its spin quantisation axis. The amplitudes from (3.31) for the case where the top quark is polarised to have spin up (↑) and spin down (↓) along the quantisation axis are given as follows [48]: Vtb bν [ t− ] t+ t−

, 2 2 ( − b) − m W + im W W m t Vtb bν [ t+ ] M(t ↓ → b + ν ) = −g22 . ( − b)2 − m 2W + im W W

M(t ↑ → b + ν ) = −g22

(3.33)

Squaring these matrix elements results in: g24 |Vtb |2 (b · ν)( · t− ) |M(t ↑ → b + ν )|2 =  , 2 2 ( − b)2 − m 2W + m 2W W g24 |Vtb |2 (b · ν)( · t+ ) |M(t → b ν )| =  . 2 2 ( − b)2 − m 2W + m 2W W ↓

Fig. 3.5 The Feynman diagram for the t → + ν b

+

(3.34)

2

t

b

W +

ν

3.3 Polarisation Phenomenology

53

The four-momenta of the various particles along with st may be given in the top rest frame with the z-direction chosen to be the quantisation axis3 , such that: st = (0, 0, 0, 1), t = (m t , 0, 0, 0),

(3.35)

= E (1, sin θ cos φ , sin θ sin φ , cos θ ). The convention is to use dimensionless variables: xi = 2E i /m t , yi =

m i2 /m 2t ,

for massless particles (3.36)

for massive particles

γ = W /m W , such that the double differential distribution is written as [112]:  m t g24 |Vtb |2 d(st ) γ x

−1   F0 (st ) tan = 3 d x d cos θ

1 + γ 2 yW 16 (4π ) yW γ

.

(3.37)

Here, the factor in front of tan−1 (·) encodes the angular dependence and is given by: " F0 (st ) =

x (1 − x )(1 + cos θ ), st = ↑, x (1 − x )(1 − cos θ ), st = ↓ .

(3.38)

Using the narrow-width approximation:

y 1 π , δ 1 − = γ →0 (1 − y/yW )2 + γ 2 γ yW lim

(3.39)

the angular distribution is obtained by integrating over x [113–115]: 1 d 1 = (1 + α cos θ ),  d cos θ

2

(3.40)

where α = ±1. The angular distributions for other decay products have been computed and have |αi | < 1 [113]. The preference to align with the t-quantisation axis can then be heuristically understood in terms of Fig. 3.6. Naïvely, the W boson may be left, right or longitudinally polarised. However, the V − A structure of the W tb vertex forces the b quark to have left-handed helicity in the massless limit. This is only consistent with a left or longitudinally polarised W . The claim follows from the conservation of angular momentum. 3 The

covariant spin vector is sμ =



|pt | E t pt m t , m t |pt |

 .

3 Probing CP -violating Top-Yukawa Couplings at the LHC

54

+ + t b

t W+

W+

ν

b

ν Fig. 3.6 Solid arrows denote physical momentum in the rest frame of the t-quark. Double-arrows denote the polarisation of the corresponding particles. With the quantisation axis of the t-quark fixed to point in the right hand side direction, one can see that the lepton is preferentially emitted along the quantisation axis. Figure based on [116]

3.3.2 Single Top Production We first consider the t-channel single top production. The reason that the produced top quark is polarised follows for the same reason as the leptonic decay due to the V − A structure of the W tb vertex. The presentation here is motivated by [117–119]. Again, the amplitude corresponding to Fig. 3.7 can be simply written down: 

 −ig2 μ γ ρ pρ + m t M = i u b Vtb √ γ P− −i 2 M p − m 2t + im t t 2      −i gμν − kμ kν /m 2W −ig2 ν × u ν √ γ P− v , k 2 − m 2W + i MW W 2 (3.41) where M = u b (k2 ) is the amplitude for the process u(k1 )b(k2 ) → t ( p)d(k3 ) with the spinor u t ( p) left out and  denoting a combination of gamma matrices. Neglecting lepton and light-quark (including b) masses and squaring gives:

Fig. 3.7 Single top production followed by leptonic decay of the top quark

q(k1 )

j(k3 )

t(p) W (k) b(k2 )

b(k4 )

3.3 Polarisation Phenomenology

|M|2 =

55

1 1 g24 |Vtv |2 2 4 ( p 2 − m 2t )2 + t2 m 2t (k 2 − m 2W )2 + W m 2W

× u b γ μ P− ( p + m t )Mu ν γμ P− v × v P+ γν u ν M ∗ γ 0 ( p + m t )P+ γ ν u b , (3.42) The completeness relation can then be used to factorise this squared amplitude: 

π  ρλλ Dλλ δ( pt2 − m 2t ), m  t t  λ λ,λ (3.43) ) where λ( specifies the helicity of the t-quark. The spin correlations in the production and decay of the t-quark are then encoded in the respective spin-density matrices4 ρλλ and Dλλ . As one expects the |M(ub → td)|2 to correspond exactly to that in (3.34) by making the substitutions → u and ν → d, the angular correlations can be extracted as follows: p − mt = 

u( pt , λ)u( pt , λ)

=⇒

|M|2 =

|M(ub → td → db + ν )|2 ∝ |M(ub → t ↑ d)|2 |M(t ↑ → + ν b)|2 + |M(ub → t ↓ d)|2 |M(t ↓ → + ν b)|2 ∝ (1 + cos θu )(1 + cos θ ) + (1 − cos θu )(1 − cos θ ) = 1 + cos θu cos θ . (3.44) This result holds even though mixed polarisation terms were neglected, due to the summation over the helicity states (cf. Eq. 3.43). There are practical considerations in employing (3.44) and specifically, reconstructing the light-quark pT within the beam makes the extraction of the cos θu term difficult [122]. This term is instead replaced with: A↑↓ :=

N↑ − N↓ , N↑ + N↓

(3.45)

which is a frame-dependent spin asymmetry that measures the degree of top polarisation by comparing the sample of tops with spin up (down) N↑(↓) . Taking into account the production process in (3.40), the angular correlation becomes: 1 1 d = (1 + A↑↓ α cos θ ).  d cos θ

2

4

(3.46)

The particular form of the decay matrix is given in [117, 119]: ρλλ = |Mλλ (t → b + ν )|2 =

2g24 |Vtv |2 (t · )(b · ν)  (k 2

2 − m 2W ) + m 2W W

 a δλλ + ˆa σλλ  ,

where the last factor is dependent on the direction of the lepton in the t-rest frame due to the Bouchiat-Michel relation [120] (cf. also [121]).

3 Probing CP -violating Top-Yukawa Couplings at the LHC

56

j(k3 )

q(k1 )

W

h(k5 )

t(k4 )

b(k2 )

j(k3 )

q(k1 )

W

b(k2 )

h(k5 )

t(k4 )

Fig. 3.8 Associated Higgs and single top production

In the spectator basis where the light jet momentum in the t-quark rest frame is chosen as the quantisation axis, a large degree of polarisation of A↑↓ ≈ 1 can be found [50].

3.4 Higgs Associated with Single Top Production at the LHC The motivation to consider Higgs-associated single top production pp → th j has been outlined in the introduction and in the papers [2, 3]. Subsequently, the advantages of such channel will be made explicit in this section, such being that (i) the cross section is enhanced with increasing ξ values, which determines the CP-phase in the top-Yukawa couplings, and (ii) the t-quark produced from such a process is polarised. Due to (ii), the angular correlation between the lepton momentum and the top quantisation axis may be used to distinguish the scalar ξ = 0 (scalar), ξ = 0.5π (pseudoscalar) and ξ = 0.25 (maximally mixed) cases. The Feynman diagrams depicting bq → th j are given in Fig. 3.8. As compared to the single top processes considered in Sect. 3.3.2, the additional Higgs is an obvious complication in the kinematic phase space. The dominant cases are when the Higgs is radiated off the W propagator or the t-quark, as the other cases will be relatively suppressed by O(m i /m t,W ). The amplitude corresponding to the left panel of Fig. 3.8 is given by (cf. App. A.5): ig24 vVtb Vud 1 1 2 − m 2 + im  k 2 − m 2 + im  4 k13 W W 24 W W W W  [t+ t− ] t− t+ [t+ 2] × t− 1 [32] − ( 14 [43] − 12 [23]) , mt m2

M(ub → t (↑) dh) =

W

ig 4 vVtb Vud 1 1 M(ub → t (↓) dh) = 2 2 − m 2 + im  k 2 − m 2 + im  4 k13 W W 24 W W W W  t+ t− [t− 2] × t+ 1 [32] − ( 14 [43] − 12 [23]) , m 2W

(3.47)

3.4 Higgs Associated with Single Top Production at the LHC

57

Fig. 3.9 Cross section of pp → th j production at 14 TeV for representative values of yt which are still allowed by the fit in [2]. With the new fit in Sect. 3.2.1, only the yt = 1.2ytS M and yt = ytS M cases survive. The shaded band denotes the region for ξ which is still allowed by the indirect constraints at 95% C.L.

whereas the one in the right panel includes the CP-violating top-Higgs coupling, and gives: 1 1 g22 yt Vtb Vud √ 2 2 2 2 k45 − m t + im t t k13 − m W + im W W 2   × [12] e−iξ ([t+ 1] 13 + [t+ 2] 23 ) − [t+ t− ]eiξ t− 3 ,

M(ub → t (↑) dh) = i

1 1 g22 yt Vtb Vud √ 2 2 2 2 k − m + im  k − m 2 t t 13 t W + im W W 45   t+ t−

× [12] e−iξ ([t− 1] 13 + [t− 2] 23 ) − m t eiξ t+ 3 . mt (3.48) Due to the interference terms and an additional Higgs competing for the phase space, it is not immediately clear (without explicit evaluation of these amplitude) as to how the CP-phase should influence the degree of polarisation. We note that packages such as [123–125] can facilitate explicit evaluation of the amplitudes in (3.47) and (3.48), but will not be pursued here. Instead, Monte Carlo generators such as [126] will be used for the calculation, which are optimised to also incorporate collider and detector effects. A first complication of studying such processes at a hadron collider is that the incoming u-quark and b-quark are supplied by pp-collisions. The protons are strongly bounded states where a perturbative treatment is not possible at low energies. This is a direct consequence of the energy scale dependence (μ) of the strong coupling constant: αs (μ2R )  2 , (3.49) αs (μ2 ) = 7 1 + 2π αs (μ2R ) ln μμ2 M(ub → t (↓) dh) = i

R

3 Probing CP -violating Top-Yukawa Couplings at the LHC

58

due to requiring physical observables to be independent of the renormalisation scale (μ R ) used to regularise the infinites of the theory (cf. Sect. 1.1). Whilst the low energy regime QCD is in the confinement phase, at energies much higher than the proton mass the strong interaction becomes asymptotically free [127, 128]. Here, the interaction between quark and gluon constituents becomes much weaker and are described by the parton model [129]. In this regime, the perturbative matrix element calculations that we have considered become applicable. However, as the participating quarks and gluons move out from the centre of collision with decreasing energy scale, they hadronise and reconfine into colourless, collimated jets. The factorisation theorem [130, 131] provides the key to describe the scattering processes at the LHC, where the hard partonic cross section σˆ , are convolved with the parton density functions (PDF) f p (xi , μ2F ) describing the momentum fraction xi of the hadron p carried by the parton i at the factorisation scale μ2F : σ ( p1 p2 → X ) =

! a,b

1

0

!

d x1 d x2 f p1 (x1 , μ2F ) f p2 (x2 , μ2F )σˆ !

1

1 |M(, μ2F )|2 . 2x x s 1 2 a,b (3.50) This cannot be computed analytically because the PDFs are given empirically and the use of Monte Carlo event generators becomes necessary. We use Madgraph-2.0.2 [126] to compute the cross section at 14 TeV and from Fig. 3.9 it is clear that σ ( pp → th j) increases with |ξ |, in agreement with [132]. The cross section was fitted to a form of: =

0

d x1 d x2

df a (x1 , μ2F ) f b (x2 , μ2F )

 2 2  σ ( pp → th j) ≈ 2.30κtS − 1.30 + 1.66κtP , σ S M ( pp → th j)

(3.51)

where the constant term is due to the W W h coupling. Such interference is not present in associated top pair production and the dependence of cross section on the couplings is then [36, 91]:  2  2 σ ( pp → tth) ≈ κtS + 0.65κtP . (3.52) σ S M ( pp → tth) An immediate consequence is that the top Yukawa couplings can then be determined up to a sign in the single top channel, relative to the gauge sector. Despite having a cross section which is O(10) higher, tth production is suppressed with increasing |ξ | such that the cross section becomes comparable to that of th j when |ξ | > 0.4π (cf. Fig. 3.10). Of course, the indirect constraints in Sect. 3.2 imply that this region is only ruled out if there are no extra particles mediating the hgg, hγ γ and h Z γ couplings. It is necessary to put bounds on the CP-phase via direct measurements of the tth coupling. It should be noted that an enhanced cross section in the th j production channel does not necessary imply CP-violation, as it could be due to a large anomalous

3.4 Higgs Associated with Single Top Production at the LHC

59

Fig. 3.10 A comparison of the pp → tth and pp → th j production cross section at 14 TeV. The width on the left (right) is due to the range of the allowed yt values based on the fit in [2] (Sect. 3.2) respectively

Yukawa coupling in the scalar scenario (i.e. yt > ytS M and ξ = 0). However, it follows from the above discussion that one should be able to resolve this ambiguity by measuring σ ( pp → th j) relative to σ ( pp → tth), given that |ξ | is sufficiently large. In either case, an enhanced cross-section will assist in the observability of the signals. We demonstrate that one can make use of the angular distribution to further distinguish the CP-phase. In collider studies, one has to also take into account that the particles described in the partonic processes do not directly map to objects that are detected at the LHC. As discussed previously, the outgoing quarks hadronises to form jets which are then detected as showers of π 0,± , p and n in the calorimeter. Even though γ , e± and μ± can be directly detected, inefficiencies are still present in their reconstruction. Finally, particles such as neutrinos interact so weakly that they can only be inferred through the momentum imbalance in the transverse plane known as missing energy E T . The result of such detector inefficiency and misidentification is that there will be an abundance of background processes with the same signatures and one has to invent a series of selection criteria to optimise observability of the signal over such backgrounds. We considered h → bb and h → γ γ separately in [2] and [3] respectively. The main backgrounds to the former decay channel are: (B1) pp → t (→ νb)t(→ b j j), which can fake the signal when one light jet from the (anti-)top quark hadronic decay is misidentified as a b-jet; (B2) pp → t (→ νb)Z (→ bb) j, which is an irreducible background5 but with a pair of b-jets coming from Z boson; (B3) pp → tbb j, which are irreducible QCD events. 5 Backgrounds are separated into two types, with reducible corresponding to those where some final

state particles fake the ones from the signal, and irreducible to those where the final states of the background and signal coincide.

3 Probing CP -violating Top-Yukawa Couplings at the LHC

60

One the other hand, with the Br (h → γ γ ) ∼ 10−3 , backgrounds of the diphoton channel are given by: (B1 ) pp → t jγ γ , which as a non-resonant irreducible process, is expected to be efficiently suppressed through a window cut on m γ γ ; (B2 ) pp → t (→ νb)t (→ bj j)γ γ , where one of the jets in the hadronically decaying top is misidentified, and another two are missed in the detector. The γ γ decay may follow from either Bremstralung radiation or decay from the Higgs; (B3 ) pp → W + (→ ν)γ γ j j, where one jet is mis-tagged as a b-jet and another missed in the detector. Again, the photon pair may result from Higgs decay.

3.4.1 Collider Physics at the LHC Due to the quantum mechanical nature of fundamental processes, the strategy employed by collider experiments is often a counting one, where the expected number of events depends on the cross section σ and the amount of data collected, measured as the integrated luminosity L: ! Nex p = σ L = σ

dt L.

(3.53)

The additional  factor is required to take into account the efficiency of the detector as well as the event selection. By the end of the HL-LHC (High Luminosity LHC) programme, it is anticipated that 3000 fb−1 of data will be collected. √ We used Madgraph-2.0.2 to simulate the signal and background events at s = 14 TeV and L = 3000 fb−1 , where parton showering and detector effects were incorporated by Pythia-6.4.28 [133] and Delphes-3.0.12 [134]. The detailed configurations to these packages can be found in our works [2, 3]. One has to understand the kinematics of the signal and appropriately choose a set of selection cuts to optimise observability of the signal. The observability is then measured in terms of the following two quantities: • signal-to-background ratio:

Nevt S = , B Nbkg

where one wants S/B > 1 for a clear observation of the signal. • signal significance: S Nevt , =# √ Nbkg + Nevt S+B

(3.54)

(3.55)

3.4 Higgs Associated with Single Top Production at the LHC

61

where one has to can quantify the likelihood that the√signal is not a fluctuation of the background. Evidence for a signal requires S/ S + B > 3, whereas for √ discovery, S/ S + B > 5. We now describe the coordinate system typically adopted by collider experiments. The convention is to choose the collision point as the origin, with the four momenta of the beams given by: √ s μ (1, 0, 0, 1), p1 = (3.56) √2 s μ p2 = (1, 0, 0, −1), 2 since the initial state partons’ transverse momenta are assumed to be negligible. We need to find variables that respect the cylindrical symmetry of the detector and that are invariant under longitudinal boosts. Given that the centre of mass for the protons (lab frame) does not coincide with that of the parton interactions in general, the momenta are often decomposed into variables that transform differently under longitudinal Lorentz boost. One uses the cylindrical symmetry of the detector and the fact that the momenta of the initial partons are negligible in the transverse plane. Specifically, the rapidity:   1  E + p L  , (3.57) y := ln  2 E−p  L

transforms additively under longitudinal boost. On the other hand, the transverse momentum: pT := ( pT cos θ, pT sin θ ) (3.58) remains invariant, and therefore is a suitable kinematic variable together with y. Since the massless approximation is valid for most of the cases, one replaces the rapidity by pseudo-rapidity: η := lim y = E→|p|

  1  1 + cos θ  ln  2 1 − cos θ 

⇒

sinh η = cot θ,

(3.59)

which basically measures the polar angle from the beam axis. The particle momenta are then usually rewritten as: μ

p =

$

pT2

cosh η + 2

m2, p

T,

pT sinh η .

(3.60)

Subsequently, an effective measure of distance within the ATLAS and CMS detectors is usually parameterised by the variable: R :=

#

(η)2 + (φ)2 .

(3.61)

62

3 Probing CP -violating Top-Yukawa Couplings at the LHC

The jets are reconstructed by what is known as the anti-k T algorithm [135], based on the distance between energy deposits i and j in the hadronic calorimeter: 2p

2p

di j := min(k T i , k T j )

(R)i2j R2

,

2p

di B := k T i ,

(3.62)

where R is a chosen parameter dictating the radius of the jet. First, the combination with minimal distance min(yi j , yi B ) is identified. One then iteratively combine i j if min(yi j , yi B ) = yi j or define i as a jet if min(yi j , yi B ) = yi B , until all objects become jets. The anti-k T algorithm chooses p = −1 in order to cluster softer jets with the hard ones before the hard jets cluster amongst themselves.

3.4.2 Observability and Lepton Forward-Backward Asymmetry We take three of the benchmark scenarios—scalar (ξ = 0), pseudoscalar (ξ = 0.5π ) and maximal admixture (ξ = 0.25π ), to study the observability of our signal. The later two cases are still inside the 95% C.L. limits of the indirect constraints obtained in [2] but not in the updated results (cf. Sect. 3.2). These are therefore important scenarios to rule out via direct measurements of tth couplings, if the SM scenario (scalar) is to be established. We keep yt = ytS M to focus on the effects of the CPphase, and subsequently comment on relaxing this assumption. The projected reach of 14 TeV LHC with 3000 fb−1 is shown in Tables 3.1 and 3.2, where the cuts (C1() ) - (C4() ) are made to enhance the sensitivity of the signals over the respective backgrounds. In the analysis, only (B1) is considered for bb decay because (B2) and (B3) are demonstrated to be insignificant in [132, 136]. Similarly, (B3 ) is neglected as the accepted cross section becomes at least an order of magnitude lower than (B1 ) and (B2 ) after a window cut on m γ γ [132, 137]. The cuts (C1) and (C1 ) are common between the analysis as they reflect the basic ATLAS and CMS detector capabilities. As is clear in Fig. 3.11, the pseudorapidity cut (C3) suppresses the background where the leading jet tends to be more central (smaller η’s) as compared to that of the signal, which tend to be more forward. On the other hand, (C2 ) is reliant on the fact the diphotons of the signals are more energetic than those from the backgrounds (cf. Fig. 3.12). Continuing, the cuts (C2) and (C3 ) impose that the b-jet6 and lepton should originate from a common top quark. The cut value of 200 GeV is chosen to be slightly larger than m t to take care of reconstruction effects. Finally, the high S/B ratio and signal significance of the diphoton channel is most likely due to the clean m γ γ reconstruction (C4 ). In comparison, the Higgs mass window cut (C4) using the m bb is significantly less effective. 6 For

the h → bb study, this is the one that minimises the Mb mass.

N Normalised

3.4 Higgs Associated with Single Top Production at the LHC

63

ξ = π/2 ξ = π/4 ξ= 0 ttmatched

0.12

0.10

0.08

0.06

0.04

0.02

0.00 -5

-4

-3

-2

-1

0

1

2

3

4

5

η

j

ξ= 0 tt γγ tj γγ

10-2

NNormalised

NNormalised

Fig. 3.11 The pseudorapidity distributions of the leading jet in the signals and backgrounds of the h → bb analysis. Source [2] 10-1

ξ= 0 tt γγ tj γγ

-2

10

10-3 10-3 10-4

10-4

10-5

10-5

0

50 100 150 200 250 300 350 400 γ

p 1 [GeV] T

0

50

100

150

200

250 γ

300

p [GeV] 2

T

Fig. 3.12 pT of the leading (left) and the subleading (right) photon in the h → γ γ analysis. Source [3]

Tables 3.1 and 3.2 show the cutflow of the bb and γ γ decay channels respectively. The optimal scenario for observation is the pure pseudoscalar case in both of these channels. However, when one takes into account that yt /ytS M ∈ [0.4, 0.6] in order to remain consistent with the indirect constraint in [2], the th j cross section will be decreased by a factor of ∼ 0.6 according to (3.51). This brings the significance in the bb channel down to ∼ 3.0σ . The significance is higher in the γ γ channel, even though an extra suppression factor of 0.94 from the decay rate brings the significance

3 Probing CP -violating Top-Yukawa Couplings at the LHC

64

Table 3.1 Cutflow of the cross sections (fb) for the signals (ξ = 0, π/4 and π/2) and the backgrounds at 14 TeV LHC. The conjugate process pp → th j has been included Cuts σ [fb] t (→ νb)h(→ bb) j t (→

νb)t (b j j) ξ =0 ξ = 0.25π ξ = 0.5π (C1)

(C2) (C3) (C4)

Ri j > 0.4, i, j = b, j or

pTb > 25 GeV, |ηb | < 2.5 pT > 25 GeV, |η | < 2.5 j |η j | < 4.7 pT > 25 GeV, Mb < 200 GeV |η j | > 2.5 |m bb − m h | < 15 GeV S/B √ S/ S + B with 3000 fb−1

0.3169

0.6700

2.1860

712.4

0.3152 0.1492 0.0443 0.0028 0.610

0.6582 0.3314 0.1102 0.0070 1.512

2.1446 1.1002 0.3762 0.0238 5.120

708.7 80.33 15.82

down to ∼ 4.3σ . We conclude that the situation remains difficult under a combined √ significance of ∼ 4.7σ . One simply anticipates S/ S + B  1σ for the pure scalar and maximally mixed cases, which are even less optimistic. If the pp → th j signal is indeed found, a study on the degree of t-polarisation can be carried out by inspecting the angular distribution of the lepton (cf. Eq. 3.46). The spin-quantisation axis is chosen to lie along the direction of the top quark in the laboratory frame. In Fig. 3.13, we see that at parton level, there is a preference for the lepton to be emitted in the top-quark direction when ξ = 0 or 0.25π , and in the anti-parallel direction when ξ = 0.5π . Whilst the scalar case is expected to preserve the downward slope when detector and reconstruction inefficiencies are taken into account, the distinguishability of the pseudoscalar and mixed scenario is almost completely washed out in the bb channel. Even though such differences between the cases is still observed in γ γ channel at reconstructed level, the slope of the ξ = 0.5π has reversed from that of the parton level. The degree of polarisation is provided in the calculated spin asymmetries in Table 3.3, defined similar to (3.45): A F B :=

σ (cos θ > 0) − σ (cos θ < 0) . σ (cos θ > 0) + σ (cos θ < 0)

(3.63)

Measuring the angle relative to the t-quantisation axis, events with cos θ > 0 are denoted as spin-up (↑), and those with cos θ < 0 are denoted as spin-down (↓). Large differences can be observed between at least two of the three CP-phases at the reconstruction level for both channels, but due to the small cross sections and reconstruction inefficiencies, we expect significances 0.4 i, j = b, j, , γ |ηb | < 2.5 pTb > 25 GeV, |η | < 2.5 pT > 25 GeV, j pT > 25 GeV, |η j | < 4.7 γ pT > 20 GeV, |ηγ | < 2.5 γ γ pT1 > 50 GeV, pT2 > 25 GeV Mb < 200 GeV |m γ γ − m h | < 5 GeV

(C2 ) (C3 ) (C4 ) S/B √ S/ S + B with 3000 fb−1

(C1 )

Cuts

4.194 4.059 3.219 0.261 1.41

4.545

9.599 9.104 6.866 0.548 2.70

10.32

σ [10−3 fb] t (→ ν b)h(→ γ γ ) j ξ =0 ξ = 0.25π

39.69 37.44 28.47 2.29 7.71

42.79

ξ = 0.5π

88.11 64.05 3.295

145.0

ttγ γ ξ =0

88.24 64.10 3.493

145.8

ξ = 0.25π

87.59 63.68 3.393

144.4

ξ = 0.5π

155.2 151.3 9.031

299.4

t jγ γ

Table 3.2 Cut flow of the cross sections for the signals and backgrounds at 14 TeV LHC. The h → γ γ contributions to the ttγ γ background are included. Conjugate processes are included here

3.4 Higgs Associated with Single Top Production at the LHC 65

NNormalised

0.18

NNormalised

3 Probing CP -violating Top-Yukawa Couplings at the LHC

66

ξ = π/2 ξ = π/4 ξ= 0

0.16 0.14

0.18

0.14

0.12

0.12

0.10

0.10

0.08

0.08

0.06

0.06

0.04

0.04

0.02 0.00 -1.0

parton level -0.5

0.02 0.0

0.5

0.00 -1.0

1.0

ξ = π/2 ξ = π/4 ξ= 0

0.16

reconstructed level -0.5

0.0

0.5

0.18

ξ = π/2 ξ = π/4 ξ= 0

0.16 0.14

0.18

0.14 0.12

0.10

0.10

0.08

0.08

0.06

0.06

0.04

0.04

0.00 -1.0

parton level -0.5

0.02 0.0

0.5

1.0

ξ = π/2 ξ = π/4 ξ= 0

0.16

0.12

0.02

1.0

cos θl NNormalised

NNormalised

cos θl

0.00 -1.0

reconstructed level -0.5

0.0

cos θl

0.5

1.0

cos θl

Fig. 3.13 The top two diagrams show the lepton forward backward asymmetry at the parton and reconstructed level for the h → bb channel at 14 TeV LHC with 3000 fb−1 of data. The corresponding figures on the bottom are for h → γ γ Table 3.3 The reconstructed-level forward-backward asymmetry A F B corresponding to Fig. 3.13 ξ

h → bb σ (cos θ > σ (cos θ < A F B (%) 0) [10−2 fb] 0) [10−2 fb]

0 0.25π 0.5π

1.458 4.687 16.81

2.080 3.991 12.76

−17.6 9.0 13.7

h → γγ σ (cos θ > σ (cos θ < A F B (%) 0) [10−4 fb] 0) [10−4 fb] 4.413 12.05 54.21

7.745 13.81 50.56

−27.40 −6.805 3.484

3.5 Remarks

67

3.5 Remarks In this chapter, we introduced the CP-violating Yukawa couplings. Since a large tth coupling of yt ∼ O(1) is expected, the focused is placed on the top-Higgs sector. The top-Yukawa couplings are parameterised in terms of a modulus yt and a phase ξ . Indirect bounds on these parameters were obtained from the production and decay rates measured by the LHC [2], and also complementarily from EDM [138]. Although pp → tth is one of the leading modes for Higgs production, we expect that pp → th j may be competitive in directly probing a CP-violating top Yukawa sector. This is because σ ( pp → th j) was found to be significantly enhanced when comparable σ ( pp → tth). Due to the large h → bb branching ratio, the observability of the pp → th j signal was investigated in [2] for ξ = 0, 0.25π and 0.5π and was found to be overwhelmed by QCD background. Although the h → γ γ channel has significantly smaller branching ratio, the enhancement due to the anomalous CPphase, and clean diphoton mass reconstruction compensates to give significantly better prospects. In the most optimistic scenario, the combination of the two modes can lead to a significance of ∼4.67σ for the pure pseudoscalar case. Observation is more difficult for the scalar and maximally mixed cases because the significance stays below 3σ . The chiral W tb coupling in the th j channel was also shown to produce a high degree of polarisation in the t-quark, which can be inferred from the decay products. We found that large differences in the spin-asymmetry makes A F B a good measure of the polarisations in both the bb and γ γ decay modes. The different CP-phases can then be distinguished, albeit with low significance.

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109. P. Motylinski, Angular correlations in t-channel single top production at the LHC. Phys. Rev. D 80, 074015 (2009). arXiv:0905.4754 110. ATLAS collaboration, G. Aad et al., Search for anomalous couplings in the W tb vertex from the measurement of double differential angular decay rates of single top quarks produced in the t-channel with the ATLAS detector, JHEP 04, 023 (2016). arXiv:1510.03764 111. CMS collaboration, V. Khachatryan et al., Measurement of top quark polarisation in t-channel single top quark production. JHEP 04, 073 (2016). arXiv:1511.02138 112. T. Nasuno, Spin correlations in top quark production at e+ e− linear colliders. PhD thesis, Hiroshima U., 1999. arXiv:hep-ph/9906252 113. M. Jezabek, J.H. Kuhn, V-A tests through leptons from polarized top quarks. Phys. Lett. B 329, 317–324 (1994). arXiv:hep-ph/9403366 114. J.H. Kuhn, How to measure the polarization of top quarks. Nucl. Phys. B 237, 77–85 (1984) 115. R.H. Dalitz, G.R. Goldstein, The decay and polarization properties of the top quark. Phys. Rev. D 45, 1531–1543 (1992) 116. T.M.P. Tait, C.P. Yuan, Single top quark production as a window to physics beyond the standard model. Phys. Rev. D 63, 014018 (2000). arXiv:hep-ph/0007298 117. S. Frixione, E. Laenen, P. Motylinski, B.R. Webber, Angular correlations of lepton pairs from vector boson and top quark decays in Monte Carlo simulations. JHEP 04, 081 (2007). arXiv:hep-ph/0702198 118. V. del Duca, E. Laenen, Top physics at the LHC. Int. J. Mod. Phys. A30, 1530063 (2015). arXiv:1510.06690 119. D. Eriksson, G. Ingelman, J. Rathsman, O. Stal, New angles on top quark decay to a charged Higgs. JHEP 01, 024 (2008). arXiv:0710.5906 120. C. Bouchiat, L. Michel, Mesure de la polarisation des electrons relativistes. Nucl. Phys. 5, 416 (1958) 121. H.E. Haber, Spin formalism and applications to new physics searches, in Spin structure in highenergy processes: Proceedings, 21st SLAC Summer Institute on Particle Physics (Stanford, CA, 1994) 26 Jul–6 Aug 1993. arXiv:hep-ph/9405376 122. G. Mahlon, Spin polarization in single top events, in Thinkshop on Top Quark Physics for Run II Batavia, Illinois, 16–18 October 1998. arXiv:hep-ph/9811219 123. J.A.M. Vermaseren, New features of FORM. arXiv:math-ph/0010025 124. H. Murayama, I. Watanabe, K. Hagiwara, K. enerugi Butsurigaku Kenkyujo (Japan), HELAS: HELicity amplitude subroutines for feynman diagram evaluations / H. Murayama, I. Watanabe and K. Hagiwara. National Laboratory for High Physics Ibaraki-ken, Japan, 1992 125. J. Kuczmarski, SpinorsExtras—Mathematica implementation of massive spinor-helicity formalism. (2014). arXiv:1406.5612 126. J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer et al., The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations. JHEP 07, 079 (2014). arXiv:1405.0301 127. D.J. Gross, F. Wilczek, Ultraviolet behavior of nonabelian gauge theories. Phys. Rev. Lett. 30, 1343–1346 (1973) 128. H.D. Politzer, Reliable perturbative results for strong interactions? Phys. Rev. Lett. 30, 1346– 1349 (1973) 129. R.P. Feynman, Very high-energy collisions of hadrons. Phys. Rev. Lett. 23, 1415–1417 (1969) 130. J.C. Collins, D.E. Soper, The theorems of perturbative QCD. Ann. Rev. Nucl. Part. Sci. 37, 383–409 (1987) 131. J.M. Campbell, J.W. Huston, W.J. Stirling, Hard interactions of quarks and gluons: A primer for LHC physics. Rept. Prog. Phys. 70, 89 (2007). arXiv:hep-ph/0611148 132. J. Chang, K. Cheung, J.S. Lee, C.-T. Lu, Probing the top-Yukawa coupling in associated Higgs production with a single top quark. JHEP 05, 062 (2014). arXiv:1403.2053 133. T. Sjöstrand, S. Ask, J.R. Christiansen, R. Corke, N. Desai, P. Ilten et al., An introduction to PYTHIA 8.2. Comput. Phys. Commun. 191, 159–177 (2015). arXiv:1410.3012 134. DELPHES 3 collaboration, J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens et al., DELPHES 3, A modular framework for fast simulation of a generic collider experiment, JHEP 02 (2014) 057. arXiv:1307.6346

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Chapter 4

Electroweak Phase Transition and Baryogenesis

Why does the whole world have φ = v? Why doesn’t it have φ = −v somewhere? —R. Feynman [1]

The observed baryon asymmetry remains an open question at the intersection of cosmology and particle physics. The latest of such measurements is given by the Planck Collaboration [2]: ηB =

nB = (8.67 ± 0.05) × 10−10 (Planck). s

(4.1)

Any physics model which possibly explains this asymmetry must satisfy the three conditions [3]: (i) B violation, (ii) C and CP violation, and (iii) a departure from thermal equilibrium (or violation of CPT invariance). It is obvious that if B = 0 initially, baryon number violating interactions are necessary for the universe to end up with a non-zero asymmetry. The second condition follows from the fact that the baryon number operator Bˆ is odd both under C and CP. Finally, if the Hamiltonian Hˆ is invariant under CPT , then the expectation of the baryon number is necessarily zero, as argued below:     ˆ ˆ B = Tr e−β H Bˆ = Tr (CPT CPT )−1 e−β H Bˆ   ˆ = Tr (CPT )−1 e−β H CPT Bˆ = −B.

(4.2)

Several mechanisms have been proposed to generate the baryon asymmetry, e.g. the Affleck-Dine mechanism [4], electroweak baryogenesis [5] and leptogenesis [6], in order to avoid significant wash-out of baryon asymmetry during the inflationary period. Particularly, electroweak baryogenesis is attractive because all the Sakharov conditions could qualitatively be realised within the SM at the electroweak phase © Springer International Publishing AG 2017 J.T.S. Yue, Higgs Properties at the LHC, Springer Theses, DOI 10.1007/978-3-319-63402-9_4

75

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4 Electroweak Phase Transition and Baryogenesis

transition. With the Higgs boson being 125 GeV, this can only be achieved via extensions of SM which are expected to be near the reach of LHC sensitivity. Section 4.1 of this chapter will explain why electroweak baryogenesis cannot be achieved in the SM alone. Non-linear realisation of the SU (2) L ⊗ U (1)Y gauge symmetry is still viable, so it is important to study its implications on electroweak baryogenesis, as is done in our work [7]. This scenario is helped by the extra source of CP-violation that was studied in Chap. 3. Additionally in Sect. 4.2, we demonstrate the associated topHiggs coupling, together with anomalous cubic term in the Higgs potential, allows for a first order phase transition. An important result is that the EW symmetry is simply not restored at high temperature. In Sect. 4.3, the bubble dynamics that occur at the phase transition will be briefly considered as it is relevant to the non-equilibrium criterion. Non-perturbative EW sphalerons are responsible for converting the chiral excess that is generated at the front of the wall, into a baryon asymmetry. It is then necessary to consider the modifications of the sphaleron rate due to the anomalous cubic couplings, as is presented in Sect. 4.4. Subsequently in Sect. 4.5, we explain how CP-violating scattering generates a net lepton number n L and diffuses in front of the bubble wall according to the transport scenario. Finally, the conversion of n L into the baryon asymmetry n B /s is addressed in Sect. 4.6 and the chapter is concluded with some remarks in Sect. 4.7.

4.1 Problems with Electroweak Baryogenesis The symmetric phase of SM Higgs field ρ = 0 is stabilised by the plasma at high temperature in the early universe. As the universe cools to temperatures near the electroweak scale T ∼ O(100 GeV), bubbles in the broken phase ρ = 0 nucleate and expand. A first-order phase transition is defined by an energy barrier which separates the true and false vacua, and gives the out-of-equilibrium condition (cf. Sect. 4.5). As the quarks in the plasma scatter off the expanding bubble wall, CP-violating processes generate a chiral excess in front of the wall. B-violating electroweak sphalerons then convert this to a baryon asymmetry which moves inside the wall. Such excess in n B persist inside the bubble, given that the sphaleron processes are suppressed there (Fig. 4.1). However, proposals to generate the baryon asymmetry during EW phase transition within the SM suffer from several deficiencies: (i) A strongly first order phase transition in the SM requires m h < 75 GeV [9–14] (ii) sphaleron suppression in the broken phase m h  35 GeV and (iii) the Cabbibo-Kobayashi-Maskawa (CKM) phase is insufficient to generate sufficient asymmetry [15–18]. This implies that extensions to the SM will be required for this mechanism to work. With precision measurements of the Higgs couplings at the LHC expected at the per-cent level, the electroweak sector will be strongly constrained. However, given that many of the Higgs couplings are known currently O(20−40%), effective field theory (EFT) provides the most direct way to revisit the viability of electroweak

4.1 Problems with Electroweak Baryogenesis

77

Fig. 4.1 A schematic of electroweak baryogenesis. Based on figure in [8]

ψR

 CP 

ψL

/ B ρ = 0 nB ρ = 0

baryogenesis. It is known that the shortcomings of the minimal Standard Model may be alleviated by two six dimensional operators [19–31]: Oh 3 =

c6 ctth (H † H )3 , Otth = 2 (H † H )Q L H t R , 2  

(4.3)

where  denotes the scale of the new physics threshold, and c6 , ctth ∼ O(1), the dimensionless Wilson coefficients. Here, the first operator acts to strengthen the first-order transition, whilst the second provides a new source for CP-violation. An immediate consequence of Oh 3 are deviations in the Higgs self couplings which may be be studied through Higgs pair production (see [30, 32–35] and references therein). However, the low cutoff  ∼ 840 GeV required for a m h = 125 GeV Higgs to generate sufficient baryon asymmetry [24] brings the validity of EFT framework into question. The simplest SM extension is that of a singlet scalar, the addition of which has been widely studied [35–44]. However, to make them viable for electroweak baryogenesis, the Higgs-singlet coupling must be sufficiently strong to change the dynamics at the phase transition. Many of these models will therefore be strongly constrained by future results from the LHC (for exceptions, cf. e.g. [35] for a ‘nightmare scenario’). Furthermore, those models relying on singlet couplings with the EW-interacting fermions for CP-violation are stringently constrained by EDM (cf. e.g. [18, 45– 47]). Otherwise these CP-violating sources may come from the CP-odd part of a complex singlet, or more complicated two Higgs doublet models [18, 48–50, 50– 54]. Although these features are realised in weak scale supersymmetry (cf. [55] for the latest discussion on model constraints), such model is far from minimal. Instead, this chapter will demonstrate that EW baryogenesis can be minimally achieved with a non-linearly realised EW gauge group.

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4 Electroweak Phase Transition and Baryogenesis

4.2 EW Phase Transition and Effective Potentials To find the vacuum expectation values in a classical theory is easy—one simply finds the Higgs configuration ρ = v := ρ that minimises the potential. However, to study the electroweak phase transition, one necessarily takes into account the quantum and thermal corrections through the effective potential. Analogous to quantum effects, we show that the thermal effects can be incorporated within the perturbation theory.

4.2.1 Tree Level Potential The non-linear realisation of the EW gauge group permits a cubic Higgs singlet term in the potential (cf. Eq. 1.57): V (ρ) = −

μ2 2 κ 3 λ 4 ρ + ρ + ρ , 2 3 4

(4.4)

where κ = 0 is necessary but not sufficient to recover the SM. However, the condition becomes sufficient if one assume that the gauge sector is left unmodified: L⊃

ρ2 (Dμ )† (D μ ). 2

(4.5)

Under this assumption, the electroweak vacuum is then identified up to a sign: g2 m 2W (v) = √ 2 4 2G F

=⇒

|v| = 246 GeV,

(4.6)

where m W can be read off (1.43) and G F is the coupling constant in Fermi’s effective theory (cf. Eq. 1.22). The sign ambiguity is to be understood as a result of the Z2 symmetry being broken by the cubic term. This can be explicitly seen from the definition of v at tree level: V (v) = −μ2 v + κv 2 + λv 3 = 0,

(4.7)

which gives two non-zero roots. In order to resolve this ambiguity, we observe that the Higgs mass is defined via the potential as: m 2h (v) := V

(v) = 3λv 2 + 2κv − μ2 = (125 GeV)2 , resulting in the following relation between parameters of the potential:

(4.8)

4.2 EW Phase Transition and Effective Potentials

79

Fig. 4.2 The classical level potential given in (4.4). One cannot recover the SM vacuum when κv < −3m 2h

 1 2 m h + vκ , 2  1  λ = 2 m 2h − vκ . 2v

μ2 =

(4.9)

After imposing λ > 0 to prevent an unstable potential, there are three cases that follow. These cases can be classified in terms of the value taken by κ (cf. Fig. 4.2): • μ2 < 0 (ρ = 0 is a minimum) which implies −3m 2h < vκ < −m 2h . The left inequality follows from V (v) < V (0). • μ2 > 0 (ρ = 0 is a maximum) which implies −m 2h < vκ < 0. The right inequality comes from demanding that V (v) ≤ V (−v) • μ2 = 0, vκ = −m 2h . In all cases, the sign of the v is opposite to that that of κ. When ρ acquires a vacuum expectation value ρ = v, the physical Higgs boson is identified as the fluctuation around this value via ρ(x) = v + h(x) as per (3.5). We are careful to note that although the κ term in (4.4) is absent in the SM, the trilinear coupling of the physical Higgs boson is in general non-zero and defined through: λ3 :=

 d 3 V  = 6λv + 2κ, =⇒ dρ3 ρ=v

λ3S M =

3m 2h . v

(4.10)

In order to acquire a strongly first order phase transition, large deviations in the self couplings are required [23, 38] and should be observable at the LHC [41].

4.2.2 One Loop Quantum Corrections In this part, we explain how one loop quantum corrections to the classical potential (4.4) are to be included. One begins with the generating functional of the n-point

80

4 Electroweak Phase Transition and Baryogenesis

(connected) correlation functions:  

Z [J ] = e

 i W [J ]



Dρe−S[ρ]+ d x J (x)ρ(x)  . −S[ρ] Dρe

:=

4

(4.11)

The definition of the classical configuration: ρc (x) := −i



δW [J ] = δ J (x)

0|ρ(x)|0 ˆ 0|0

,

(4.12)

J

then corresponds to the vacuum expectation of the field operator in the presence of the source J . From this, the effective action is related to the generator for the connected Greens function via:

 [ρc ] := min W [J ] − d 4 x J (x)ρc (x) . (4.13) The effective action is name as such because at tree level, [ρc ] gives the full quantum theory with S[ρ]: e

 i[ρc ]+ d 4 x J ρc

=e

i W [J ]

 =



Dρei S[ρ]+

d4 x J ρ

.

(4.14)

In fact, one can show that:  [ρc ] = S[ρc ] − i  ln

Dξ exp

i 1 2



 d4x

d4 y

  δ 2 S[ρ] , ξ(x)ξ(y) + O ξ 3 δρ(x)δρ(y)

(4.15) where the second term includes the one-loop quantum corrections. The effective potential will now include Coleman-Weinberg terms: V (ρ) = V (0) (ρ) + VC(1) W (ρ) + · · · ,

(4.16)

since it is identified as the zeroth order term of the effective action expansion in  momentum space [ρ] = − d 4 x V . The Coleman-Weinberg contribution then takes the explicit form: ⎡ (1)

VC W (ρ) = −

⎛

⎢ ⎜1 1 ⎢ ⎜ (−1)2si (1 + 2si ) ⎢m i4 (ρ) ⎜ − γ E + ln(4π) − ln ⎣ ⎝ 64π 2   i



m i2 (ρ) μ2R



⎞⎤ ⎟⎥ ⎟⎥ + C s ⎟⎥ , ⎠⎦

:=CU V

(4.17) where Cs = 23 for all species in the DR scheme, as all Lorentz contractions of momenta are kept in 4 dimensions [56, 57]. We will only be concerned with the

4.2 EW Phase Transition and Effective Potentials

81

t-quark, the Higgs boson and the vector bosons since their large masses make them the dominant contributions. It is simple to calculate their effective masses (as the coefficient of quadratic fluctuation of each field after expanding the Lagrangian around the background field(s) [23]), as well as the respective number of degrees of freedom which determine the multiplicities of their contributions in (4.17): n h = 1,

m 2h (ρ) = 3λρ2 + 2κρ − μ2 , g22 + g12 2 ρ , 4 g2 m 2W (ρ) = 2 ρ2 , 4

2

2 ! yt yt 2

m t + √ ρ cos ξ + √ ρ sin ξ m t (ρ) = . 2 2 (4.18)

n Z = 3,

m 2Z (ρ) =

n W = 6, n t = −12,

However, to find the effective m 2t (ρ) with CP-violating top-Yukawa interaction, it is more instructive to examine the origins of (4.17). This is given by the two derivative term in (4.15) which is evaluated as: −i (−1)2s V (1) (ρ) = 2



⎡ ⎤ d4 p ln det ⎣iγ0 p0 + γi pi − (yS + m + iγ 5 y P )⎦ .    (2π)4 :=K

(4.19) For the SM scenario, yS =

yf ρ √ 2

and y P = 0, so one can simply use the fact that [58]:

Tr ln K = ln det K and Tr ln(γ μ pμ − m) =

1 Tr ln(m 2 − p 2 ), 2

(4.20)

to show that fermion cases coincide with the boson cases. For the CP-violating top y ρ y ρ Yukawa coupling considered in this work, yS = √f 2 cos ξ and y P = √f 2 sin ξ. It was shown in [59], that multiplication of K by γ0 reshuffles the i p0 ’s onto the diagonal, but leaves the determinant unaltered. The determinant is then: det K =

4 " (λi − i p0 ),

(4.21)

i=1

where λi are the eigenvalues of γ 0 K when p0 = 0. The symmetries of#the inverse propagator ensure that half of the eigenvalues are each E p = ± p2 + (yS + m)2 + y 2P . We discuss how to treat thermal corrections in the next part.

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4 Electroweak Phase Transition and Baryogenesis

4.2.3 Finite Temperature Corrections In this part, it will be argued that thermal effects may be included by compactifying the imaginary time axis. One has to first consider the expectation value of an observable Aˆ in a thermal bath of temperature T , which is given by:  

−β E n ˆ n| A|ne , Aβ := Tr ρ Aˆ = Z −1

(4.22)

n

where |n is the eigenstate corresponding energy E n under Hˆ . The density operator ρ (not to be confused with the singlet scalar field) is defined in terms of the partition function Z : 1 ˆ ˆ Z = Tr e−β H . (4.23) ρ := e−β H , Z At the path-integral level, the in- and out- states are identical due to the trace in (4.22). Given also the extra thermal e−β En function, the integral follows a thermal contour C that begins at t and ends at t − iβ. A formal equivalence between quantum statistical mechanics and quantum field theory can then be made [60]: t := iτ = iβ

=⇒

ˆ

ˆ

e−β H = ei H t/ ,

(4.24)

such that thermal averaging via the exponential of the inverse temperature corresponds to imaginary time evolution. Subsequent information about the thermal system are encoded in the finite temperature n-point Green function, which can be obtained by acting on (4.23) using n derivatives: G (n) (x1 , . . . , xn ) := TC (ρ(x1 ) . . . ρ(xn )),

(4.25)

with TC being the time-ordering operator along the path C. Specifically, the two point function is important as relate to the propagators of the Feynman diagrammatic rules. Assuming that convergence is controlled by exponentials, (4.22) implies that one can define G + (x, y) on −β < Im(x0 − y0 ) < 0 and G − (x, y) on 0 < Im(x0 − y0 ) < β such that: ˆ ˆ ρ(y). (4.26) G + (x, y) = G − (y, x) := ρ(x) The thermal propagator can now be defined via these two-point functions: G(x − y) = θC (x 0 − y 0 )G + (x, y) + θC (y 0 − x 0 )G − (x, y),

(4.27)

such that it is analytic on −β < Im(x0 − y0 ) < β. This condition implies a path of decreasing imaginary time, and is sufficient for the existence of all n-point Greens function [61].

4.2 EW Phase Transition and Effective Potentials

83

The formulation of a quantum field theory is specific to the contour chosen [62], with the simplest choice being a straight line along the imaginary time axis [63]. Under such imaginary time or Matusbara prescription (4.24), the Feynman rules for the finite temperature differs from the vacuum version in that the imaginary time axis is compactified. A brief description will be given below but the readers are referred to e.g. [64–67] for more pedagogical discussions. Starting with −iτ = x 0 − y 0 , the two-point correlator from (4.27) can be rewritten into the form:  $ % d4 p ρ( p)e−ωτ eip·(x−y) θ(τ ) + (−1)2s n(ω) , (4.28) G(τ , x − y) = 4 (2π) where the spectral function is given by ρ( p) := 2π [θ(ω) − θ(−ω)] δ( p 2 − m 2 ) and the thermal distribution is either Fermi-Dirac or Bose-Einstein according to the spin s: 1 . (4.29) n(ω) = βω e − (−1)2s Subsequently, one notes that the thermal Greens functions are (anti-)periodic1 : G(τ + β) = (−1)2s G(τ ),

−β < τ